We partially answer two questions of Goodaire by showing that in a finite, strongly right alternative ring, the set of units (if the ring is with unity) is a Bol loop under ring multiplication, and the set of quasiregular elements is a Bol loop under "circle" multiplication.
The supercommutator algebra of a right alternative superalgebra is a Bol superalgebra. Hom-Bol superalgebras are defined and it is shown that they are closed under even self-morphisms. Any Bol superalgebra along with any even self-morphism is twisted into a Hom-Bol superalgebra. The supercommutator algebra of a right Hom-alternative superalgebra has a natural Hom-Bol superalgebra structure.
Every multiplicative Hom-Malcev algebra has a natural multiplicative Hom-Lie triple system structure. Moreover, there is a natural Hom-Bol algebra structure on every multiplicative Hom-Malcev algebra and on every multiplicative right (or left) Hom-alternative algebra.
In this paper, we study rings having the property that every right ideal is automorphism-invariant. Such rings are called right $a$-rings. It is shown that (1) a right $a$-ring is a direct sum of a square-full semisimple artinian ring and a right square-free ring, (2) a ring $R$ is semisimple artinian if and only if the matrix ring $\mathbb{M}_n(R)$ for some $n>1$ is a right $a$-ring, (3) every right $a$-ring is stably-finite, (4) a right $a$-ring is von Neumann regular if and only if it is semiprime, and (5) a prime right $a$-ring is simple artinian. We also describe the structure of an indec…
It shown that the supercommutator superalgebra of a right alternative superalgebra is a Bol superalgebra. Hom-Bol superalgebras are defined and it is shown that they are closed under even self-morphisms. Any Bol superalgebra along with any even self-morphism is twisted into a Hom-Bol superalgebra. The supercommutator superalgebra of a right Hom-alternative superalgebra has a natural Hom-Bol structure. In order to prove this last result, the Hom-Jordan-admissibility of right Hom-alternative superalgebras is investigated and next Hom-Jordan supertriple systems are defined and their connection wi…
If $A$ is a ring with automorphism $φ$ and the skew Laurent series ring $A((x,φ))$ is a right semidistributive semilocal ring then $A$ is a right semidistributive right Artinian ring. The Laurent series ring $A((x))$ is a right semidistributive semilocal ring if and only if $A$ is a right semidistributive right Artinian ring. The study is supported by Russian Scientific Foundation (project 16-11-10013).
Let $A$ be a ring and $φ$ its automorphism. It is proved that skew Laurent series ring $A((x,φ))$ is a right serial ring if and only if $A$ is a right serial right Artinian ring.
A ring $R$ is said to be right McCoy if the equation $f(x)g(x)=0,$ where $f(x)$ and $g(x)$ are nonzero polynomials of $R[x],$ implies that there exists nonzero $s \in R$ such that $f(x)s = 0$. It is proven that no proper (triangular) matrix ring is one-sided McCoy. If there exists the classical right quotient ring $Q$ of a ring $R$, then $R$ is right McCoy if and only if $Q$ is right McCoy. It is shown that for many polynomial extensions, a ring $R$ is right McCoy if and only if the polynomial extension over $R$ is right McCoy. Other basic extensions of right McCoy rings are also studied.\left…
Regarding the question of how idempotent elements affect reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce {\it right} (resp., {\it left}) {\it $e$-reversible rings}. We show that this concept is not left-right symmetric. Basic properties of right $e$-reversibility in a ring are provided. Among others it is proved that if $R$ is a semiprime ring, then $R$ is right $e$-reversible if and only if it is right $e$-reduced if and only if it is $e$-symmetric if and only if it is right $e$-semicommutative. Also, for a right $e…
Let $A$ be a ring with minimum condition on principal right ideals. It is proved that $\aleph_0$-distributive right (left) $A$-modules coincide with Artinian (Noetherian) right (left) $A$-modules. Rings, over which all right modules are direct sums of $\aleph_0$-distributive coincide with rings of of finite representation type. Rings, whose right modules are semidistributive, coincide with Kawada rings, over basis rings of which all right modules are completely cyclic. The studies of Tuganbaev are supported by Russian Scientific Foundation, project 22-11-00052.
We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations t…
$\textbf{Theorem 1.3.}$ For a given ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ Every non-singular right $A$-module $X$ which is is injective with respect to some essential right ideal of the ring $A$ is an injective module. $\textbf{2)}$ $A/G(A_A)$ is a right strongly semiprime ring. $\textbf{Theorem 1.4.}$ For a given ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right strongly semiprime ring. $\textbf{2)}$ Every right $A$-module which is injective with respect to some essential right ideal of the ring $A$, i…
$\textbf{Theorem 1.2.}$ For a ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right automorphism-invariant right non-singular ring. $\textbf{2)}$ $A$ is a right automorphism-invariant regular ring. $\textbf{3)}$ $A=S\times T$, where $S$ is a right injective regular ring and $T$ is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. $\textbf{Theorem 1.5.}$ For a ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ $A/G(A_A)$ is a semiprime right Goldie ring. $\textbf{2)}$ …
This paper presents the right-left symmetry of the CS and max-min CS conditions on nonsingular rings, and generalization to nonsingular modules. We prove that a ring is right nonsingular right CS and left Utumi if and only if it is left nonsingular left CS and right Utumi. A nonsingular Utumi ring is right max (resp. right min, right max-min) CS if and only if it is left min (resp. left max, left max-min) CS. In addition, a semiprime nonsingular ring is right max-min CS with finite right uniform dimension if and only if it is left max-min CS with finite left uniform dimension.
Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $ω: S\rightarrow End(R)$ a monoid homomorphism. In this paper we study the ascending chain conditions on principal left (resp. right) ideals of the skew generalized power series ring $R[[S,ω]]$. Among other results, it is shown that $R[[S,ω]]$ is a right archimedean reduced ring if $S$ is an Artinian strictly totally ordered monoid, $R$ is a right archimedean and $S$-rigid ring which satisfies the ACC on annihilators and $ω_s$ preserves nonunits of $R$ for each $s\in S$. As a consequence we deduce that the power series rings, Laurent …
Let $R$ be a ring. $R$ is called a right countably $Σ$-C2 ring if every countable direct sum copies of $R_{R}$ is a C2 module. The following are equivalent for a ring $R$: (1) $R$ is a right countably $Σ$-C2 ring. (2) The column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_{\mathbb{N}}(R)$ is a right C2 (or C3) ring. (3) Every countable direct sum copies of $R_{R}$ is a C3 module. (4) Every projective right $R$-module is a C2 (or C3) module. (5) $R$ is a right perfect ring and every finite direct sum copies of $R_{R}$ is a C2 (or C3) module. This shows that right countably $Σ$-C2 rings a…
We show that practically all the properties of almost perfect rings discovered by Bazzoni and Salce in "Almost perfect domains" (Colloq. Math. 95 (2) (2003), 285-301) for commutative rings hold in the non-commutative setting.
Perfect Gabriel filters of right ideals and their corresponding right rings of quotients have the desirable feature that every module of quotients is determined solely by the right ring of quotients. On the other hand, symmetric rings of quotients have a symmetry that mimics the commutative case. In this paper, we study rings of quotients that combine these two desirable properties. We define the symmetric versions of a right perfect ring of quotients and a right perfect Gabriel filter -- the perfect symmetric ring of quotients and the perfect symmetric Gabriel filter and study their propertie…
Let R be a commutative ring with unity $1\in R$. In this article, we introduce the concept of prime principal right ideal rings (\textbf{PPRIR}), A prime ideal P of R is said to be prime principal right ideal (\textbf{PPRI}) is given by $P =\{ ar : r\in R\}$ for some element a. The ring R is said to be prime principal right ideal ring (\textbf{PPRIR}) if every prime ideal of R is a prime principal right ideal (\textbf{PPRI}). A prime principal right ideal ring R is called a prime principal right ideal domain (\textbf{PPRID}) if R is a domain. Several properties and characteristics of prime pri…
R is called a right WV -ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV -ring, then R is right uniform or a right V -ring. It is shown that for a right WV-ring R, R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS or noetherian module. For a finitely generated module M with projective socle over a V -ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X \oplus T, where X is semisimple and T is noetherian with zero socle. I…
Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these right ideals and their commutative counterparts. We prove the Completely Prime Ideal Principle, a theorem stating that right ideals that are maximal in a specific sense must be completely prime. We offer a number of applications of the Completely Prime Ideal Principle arising from many diverse concepts in rings and modules. These applications show how comple…
The total right ring of quotients $Q_{\mathrm{tot}}^r(R),$ sometimes also called the maximal flat epimorphic right ring of quotients or right flat epimorphic hull, is usually obtained as a directed union of a certain family of extension of the base ring $R$. In [K. Morita, Flat modules, injective modules and quotient rings, Math. Z. 120 (1971) 25--40], $Q_{\mathrm{tot}}^r(R)$ is constructed in a different way, by transfinite induction on ordinals. Starting with the maximal right ring of quotients $Q_{\mathrm{max}}^r(R)$, its subrings are constructed until $Q_{\mathrm{tot}}^r(R)$ is obtained. H…
A ring $R$ is called right (small) dual if every (small) right ideal of $R$ is a right annihilator. Left (small) dual rings can be defined similarly. And a ring $R$ is called (small) dual if $R$ is left and right (small) dual. It is proved that $R$ is a dual ring if and only if $R$ is a semilocal and small dual ring. Several known results are generalized and properties of small dual rings are explored. As applications, some characterizations of QF rings are obtained through small dualities of rings.
The rings whose simple right modules are absolutely pure are called right $SAP$-rings. We give a new characterization of right $SAP$ rings, right $V$ rings, and von Neumann regular rings. We also obtain a new decomposition theory of right selfinjective von Neumann regular rings. The relationships between $SAP$-rings, $V$-rings, and von Neumann regular rings are explored. Some recent results obtained by Faith are generalized and the results of Wu-Xia are strengthened.
Let $R$ be a ring, a right ideal $I$ of $R$ is called small if for every proper right ideal $K$ of $R$, $I+K\neq R$. A ring $R$ is called right small injective if every homomorphism from a small right ideal to $R_{R}$ can be extended to an $R$-homomorphism from $R_{R}$ to $R_{R}$. Properties of small injective rings are explored and several new characterizations are given for $QF$ rings and $PF$ rings, respectively.
In this paper a simple right R-module S over a ring R is called hypersimple if its injective hull E(S) is cyclic, and a ring R is called right hypersimple if every simple right R-module is hypersimple. We initiate a study of these new notions, and revisit Osofsky's work on hypercyclic rings, i.e. rings whose cyclic right modules have cyclic injective hulls.
A ring $R$ is called right SSP (SIP) if the sum (intersection) of any two direct summands of $R_{R}$ is also a direct summand. Left sides can be defined similarly. The following are equivalent: (1) $R$ is right SSP. (2) $R$ is right C3 and right SIP. (3) $R$ is left C3 and left SIP. (4) $R$ is left SSP. It is also shown that (1) $R$ is a von-Neumann regular ring if and only if $\mathbb{M}_{2}(R)$ is right SSP if and only if $\mathbb{M}_{n}(R)$ is right SSP for some $n>1$; (2) $R$ is a semisimple ring if and only if the column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_Λ(R)$ is right SS…
It is proved that a ring $R$ is a right uniserial, right Noetherian centrally essential ring if and only if $R$ is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings. Victor Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. Askar Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.
We introduce Central McCoy rings, which are a generalization of McCoy rings and investigate their properties. For a ring R, we prove that R is right Central McCoy if and only if the polynomial ring R[x] is right Central McCoy. Also, we give some examples to show that if R is right Central McCoy, then Mn(R) and Tn(R) are not necessary right Central McCoy, but Dn(R) and Vn(R) are right Central McCoy, where Dn(R) and Vn(R) are the subrings of the triangular matrices with constant main diagonal and constant main diagonals, respectively.
It is well known that a ring $R$ is right Kasch if each simple right $R$-module embeds in a projective right $R$-module. In this paper we study the dual notion and call a ring $R$ right dual Kasch if each simple right $R$-module is a homomorphic image of an injective right $R$-module. We prove that $R$ is right dual Kasch if and only if every finitely generated projective right $R$-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if…
A right $R$-module $M$ is called max-projective provided that each homomorphism $f:M \to R/I$ where $I$ is any maximal right ideal, factors through the canonical projection $π: R \to R/I$. We call a ring $R$ right almost-$QF$ (resp. right max-$QF$) if every injective right $R$-module is $R$-projective (resp. max-projective). This paper attempts to understand the class of right almost-$QF$ (resp. right max-$QF$) rings. Among other results, we prove that a right Hereditary right Noetherian ring $R$ is right almost-$QF$ if and only if $R$ is right max-$QF$ if and only if $R=S\times T$ , where $S$…
In recent years, centrally essential rings have been intensively studied in ring theory. In particular, they find applications in homological algebra, group rings, and the structural theory of rings. The class of essentially central rings strongly extends the class of commutative rings. For such rings, a number of recent papers contain positive answers to some important questions from ring theory that previously had positive answers for commutative rings and negative answers in the general case. This work is devoted to a similar topic. A familiar description of right Noetherian, right distribu…
We consider some existing results regarding rings for which the classes of torsion-free and non-singular right modules coincide. Here, a right $R$-module $M$ is non-singular if $xI$ is nonzero for every nonzero $x \in M$ and every essential right ideal $I$ of $R$, and a right $R$-module $M$ is torsion-free if $Tor_{1}^{R}(M,R / Rr)=0$ for every $r \in R$. In particular, we consider a ring $R$ for which the classes of torsion-free and non-singular right $S$-modules coincide for every ring $S$ Morita-equivalent to $R$. We make use of these results, as well as the existence of a Morita-equivalenc…
It is proved that the ring $R$ with center $Z(R)$, such that the module $R_{Z(R)}$ is an essential extension of the module $Z(R)_{Z(R)}$, is not necessarily right quasi-invariant, i.e., maximal right ideals of the ring $R$ are not necessarily ideals. We use central essentiality to obtain conditions which are sufficient to the property that all maximal right ideals are ideals
Let $R$ be a right and left Ore ring, $S$ its set of regular elements and $Q = R[S^{-1}] = [S^{-1}] R$ the classical ring of quotients of $R$. We prove that if F.dim$(Q_Q) = 0$, then the following conditions are equivalent: $(i)$ Flat right $R$-modules are strongly flat. $ (ii)$ Matlis-cotorsion right $R$-modules are Enochs-cotorsion. $(iii) $ $h$-divisible right $R$-modules are weak-injective. $(iv)$ Homomorphic images of weak-injective right $R$-modules are weak-injective. $(v)$ Homomorphic images of injective right $R$-modules are weak-injective. $(vi)$ Right $R$-modules of weak dimension $…
This paper studies similarities and differences between the classes of rings over which each simple module is injective and rings over which each simple module is $Σ$-injective. The rings in the former class are called $V$-rings and the rings in the latter class are called $Σ$-$V$ rings. We have obtained analogues of various well-known results about $V$-rings for $Σ$-$V$ rings. Motivated by a conjecture of Kaplansky, Fisher asked if a prime right $V$-ring is right primitive. Although a counter-example to Kaplansky's conjecture was constructed long ago but Fisher's question is still open. In th…
The Faith-Menal conjecture says that every strongly right $Johns$ ring is $QF$. The conjecture is also equivalent to say every right noetherian left $FP$-injective ring is $QF$. In this short article, we show that the conjecture is true under the condition(a proper generalization of left $CS$ condition)that every nonzero complement left ideal is not small(a left ideal $I$ is called small if for every left ideal $K$, $K$+$I$=$R$ implies $K$=$R$). It is also proved that (1) $R$ is $QF$ if and only if $R$ is a left and right mininjective ring with $ACC$ on right annihilators in which $S_{r}\subse…
Two elements $a,b$ in a ring $R$ form a right coprime pair, written $\langle a,b\rangle$, if $aR+bR=R$. Right coprime pairs have shown to be quite useful in the study of left cotorsion or exchange rings. In this paper, we define the class of strongly right exchange rings in terms of descending chains of them. We show that they are semiregular and that this class of rings contains left injective, left pure-injective, left cotorsion, local and left continuous rings. This allows us to give a unified study of all these classes of rings in terms of the behaviour of descending chains of right coprim…
We define when a noetherian ring R is called a right ( or a left) weakly krull symmetric ring . We then prove that if R is a right ( or a left ) krull homogenous ring then R is a right ( or a left ) weakly krull symmetric ring . This result modifies the main result of [2] . The key terms introduced in this paper are of independent interest .
A characterization of right (left) quasi-duo skew polynomial rings of endomorphism type and skew Laurent polynomial rings are given. In particular, it is shown that (1) the polynomial ring R[x] is right quasi-duo iff R[x] is commutative modulo its Jacobson radical iff R[x] is left quasi-duo, (2) the skew Laurent polynomial ring is right quasi-duo iff it is left quasi-duo. These extend some known results concerning a description of quasi-duo polynomial rings and give a partial answer to the question posed by Lam and Dugas whether right quasi-duo rings are left quasi-duo.
We introduce the notion of a severe right Ore set in the main as a tool to study universal localisations of rings but also to provide a short proof of P. M. Cohn's classification of homomorphisms from a ring to a division ring. We prove that the category of finitely presented modules over a universal localisation is equivalent to a localisation at a severe right Ore set of the category of finitely presented modules over the original ring. This allows us to describe the structure of finitely presented modules over the universal localisation as modules over the original ring.
In this paper we study right $S$-Noetherian rings and modules, extending of notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right $S$-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain $S$-Noetherian modules with the following consequence in commutative algebra: If a module $M$ over a commutative ring is $S$-Noetherian with respect to a multiplicative set $S$ that contains no zero-divisors for $…
It was proved independently by both Wolfson [An ideal theoretic characterization of the ring of all linear transformations, Amer. J. Math. 75 (1953), 358-386] and Zelinsky [Every Linear Transformation is Sum of Nonsingular Ones, Proc. Amer. Math. Soc. 5 (1954), 627-630] that every linear transformation of a vector space $V$ over a division ring $D$ is the sum of two invertible linear transformations except when $V$ is one-dimensional over $\mathbb Z_2$. This was extended by Khurana and Srivastava [Right self-injective rings in which each element is sum of two units, J. Algebra and its Appl., V…
In this paper we introduce the definition of a noetherian disjoint ring and that of a noetherian non-disjoint ring . For a noetherian ring R , with nilradical N if P and Q represent the semiprime ideals of R called as the right and the left krull-homogenous parts of N as defined in [8] , then we prove the main theorem of this paper for the ring R whose statement is given below. Main Theorem :- Let R be a Noetherian ring with nilradical N . Let P and Q represent the right and the left krull-homogenous parts of N . Then the following hold true for the ring R ; (a) If R is a disjoint ring , then …
A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contain a weak basis. In the paper we study (1) rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, (2) regularly weakly based modules over Dedekind domains.
In \cite{F}, Faith asked for what rings $R$ does the Dual Baer Criterion hold in Mod-$R$, that is, when does $R$-projectivity imply projectivity for all right $R$-modules? Such rings $R$ were called right testing. Sandomierski proved that if $R$ is right perfect, then $R$ is right testing. Puninski et al.\ \cite{AIPY} have recently shown for a number of non-right perfect rings that they are not right testing, and noticed that \cite{T2} proved consistency with ZFC of the statement {\lq}each right testing ring is right perfect{\rq} (the proof used Shelah's uniformization). Here, we prove the com…
Given a ring R, we define its right i-profile (resp. right p-profile) to be the collection of injectivity domains (resp. projectivity domains) of its right R-modules. We study the lattice theoretic properties of these profiles and consider ways in which properties of the profiles may determine the structure of rings and viceversa. We show that the i-profile is isomorphic to an interval of the lattice of linear filters of right ideals of R, and is therefore modular and coatomic. In particular, we give a practical characterization of the i-profile of a right artinian ring. We show through an exa…
We observe that the class of left and right artinian left and right morphic rings agrees with the class of artinian principal ideal rings. For $R$ an artinian principal ideal ring and $G$ a group, we characterize when $RG$ is a principal ideal ring; for finite groups $G$, this characterizes when $RG$ is a left and right morphic ring. This extends work of Passman, Sehgal and Fisher in the case when $R$ is a field, and work of Chen, Li, and Zhou on morphic group rings.
A ring $R$ is called a J-regular ring if R/J(R) is von Neumann regular, where J(R) is the Jacobson radical of R. It is proved that if R is J-regular, then (i) R is right n-injective if and only if every homomorphism from an $n$-generated small right ideal of $R$ to $R_{R}$ can be extended to one from $R_{R}$ to $R_{R}$; (ii) R is right FP-injective if and only if R is right (J, R)-FP-injective. Some known results are improved.
It has been shown by McCoy that a right ideal of a polynomial ring with several indeterminates has a non-trivial homogeneous right annihilator of degree 0 provided its right annihilator is non-trivial to begin with. In this note, it is documented that any $\mathbb{N}$-graded ring $R$ has a slightly weaker property: the right annihilator of a right ideal contains a homogeneous non-zero element, if it is non-trivial to begin with. If $R$ is a subring of a $\mathbb{Z}^k$ -graded ring $S$ satisfying a certain non-annihilation property (which is the case if $S$ is strongly graded, for example), the…
In this paper, we introduce and study the concept of CS-Rickart modules, that is a module analogue of the concept of ACS rings. A ring $R$ is called a right weakly semihereditary ring if every its finitly generated right ideal is of the form $P\oplus S,$ where $P_R$ is a projective module and $S_R$ is a singular module. We describe the ring $R$ over which $\mathrm{Mat}_n (R)$ is a right ACS ring for any $n \in \mathbb {N}$. We show that every finitely generated projective right $R$-module will to be a CS-Rickart module, is precisely when $R$ is a right weakly semihereditary ring. Also, we prov…
Let $R$ be a ring and $σ$ an endomorphism of $R$. In this note, we study skew polynomial rings and skew power series rings over idempotent reflexive rings and abelian rings. Also, we introduce the concept of right (resp., left) $σ$-idempotent reflexive rings which generalizes right (resp., left) idempotent reflexive rings and $σ$-abelian rings. Certain results are obtained as corollaries from our results.
A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and nonsingular module is almost clean and that every right CS and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including finite $AW^*$-algebras and noetherian Leavitt path algebras in particular, are almost clean. We say that a ring $R$ is special clean (special almost clean) if each element $a$ can be decomposed …
