We study in detail the %Shrek algebra $\mS_n$ in the title which is an algebra obtained from a polynomial algebra $P_n$ in $n$ variables by adding commuting, {\em left} (but not two-sided) inverses of the canonical generators of $P_n$. The algebra $\mS_n$ is non-commutative and neither left nor right Noetherian but the set of its ideals satisfies the a.c.c., and the ideals {\em commute}. It is proved that the classical Krull dimension of $\mS_n$ is $2n$; but the weak and the global dimensions of $\mS_n$ are $n$. The prime and maximal spectra of $\mS_n$ are found, and the simple $\mS_n$-modules are classified. It is proved that the algebra $\mS_n$ is central, prime, and {\em catenary}. The set $\mI_n$ of idempotent ideals of $\mS_n$ is found explicitly. The set $\mI_n$ is a finite distributive lattice and the number of elements in the set $\mI_n$ is equal to the {\em Dedekind} number $\gd_n$.
Explicit generators are found for the group of automorphisms of the algebra of one-sided inverses of a polynomial algebra in $n$ variables. An analogue of the polynomial Jacobian homomorphism is found.
In this paper we introduce a new generalized inverse in a ring -- one-sided $(b, c)$-inverse, derived as an extension of $(b, c)$-inverse. This inverse also generalizes one-sided inverse along an element, which was recently introduced by H. H. Zhu et al. [H. H. Zhu, J. L. Chen, P. Patrício, Further results on the inverse along an element in semigroups and rings, Linear Multilinear Algebra, 64 (3) (2016) 393-403]. Also, here we present one-sided annihilator $(b, c)$-inverse, which is an extension of the annihilator $(b, c)$-inverse. Necessary and sufficient conditions for the existence of these…
In this article one-sided (b, c)-inverses of arbitrary matrices as well as one-sided inverses along a (not necessarily square) matrix, will be studied. In adddition, the (b, c)-inverse and the inverse along an element will be also researched in the context of rectangular matrices.
Explicit generators are found for the group $G_2$ of automorphisms of the algebra $\mS_2$ of one-sided inverses of a polynomial algebra in two variables over a field of characteristic zero. Moreover, it is proved that $$ G_2\simeq S_2\ltimes \mT^2\ltimes \Z\ltimes ((K^*\ltimes E_\infty (\mS_1))\boxtimes_{\GL_\infty (K)}(K^*\ltimes E_\infty (\mS_1)))$$ where $S_2$ is the symmetric group, $\mT^2$ is the 2-dimensional torus, $E_\infty (\mS_1)$ is the subgroup of $\GL_\infty (\mS_1)$ generated by the elementary matrices. In the proof, we use and prove several results on the index of operators, and…
An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mS_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, ..., x_n]$ over an arbitrary ring $A$: $$ \lgldim (\mS_n(A))= \lgldim (A) +n.$$ The algebra $\mS_n(A)$ is noncommutative, neither left nor right Noetherian and not a domain. The proof is based on a generalization of the Theorem of Kaplansky (on the projective dimension) obtained in the paper. As a consequence it is proved that for a left or right Noetherian algebra $A$: $$ \wdim (\mS_n(A))= \wdim (A) +n.$$
In this paper, we introduce the notion of a (generalized) right core inverse and give its characterizations and expressions. Then, we provide the relation schema of (one-sided) core inverses, (one-sided) pseudo core inverses and EP elements.
The algebra $\mS_n$ in the title is obtained from a polynomial algebra $P_n$ in $n$ variables by adding commuting, {\em left} (but not two-sided) inverses of the canonical generators of $P_n$. Ignoring non-Noetherian property, the algebra $\mS_n$ belongs to a family of algebras like the Weyl algebra $A_n$ and the polynomial algebra $P_{2n}$. The group of automorphisms $G_n$ of the algebra $\mS_n$ is found: $$ G_n=S_n\ltimes \mT^n\ltimes \Inn (\mS_n) \supseteq S_n\ltimes \mT^n\ltimes \underbrace{\GL_\infty (K)\ltimes... \ltimes \GL_\infty (K)}_{2^n-1 {\rm times}}=:G_n' $$ where $S_n$ is the sym…
The classes of 1MP-inverses and MP1-inverses are recently introduced classes of generalized inverses of complex matrix. Actually, they coincide with the classes of $\{1,2,3\}$ and $\{1,2,4\}$ inverses, respectively. We consider these inverses in the context of a ring with involution and prove that their most important characterizations and properties remain true. We show that the binary relations based on these inverses are in fact the well known left-star and right-star partial orders. We extend these relations to the ring case, connect them with the unified theory of partial order relations …
We build on the description of left congruences on an inverse semigroup in terms of the kernel and trace due to Petrich and Rankin. The notion of an inverse kernel for a left congruence is developed. Various properties of both the trace and inverse-kernel are discussed, in particular that both the trace and inverse-kernel maps are onto $\cap$-homomorphisms. The lattice of left congruences is identified as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. We use this to describe the lattice of left congruences on the b…
The algebra $\mS_n$ of one-sided inverses of a polynomial algebra $P_n$ in $n$ variables is obtained from $P_n$ by adding commuting, {\em left} (but not two-sided) inverses of the canonical generators of the algebra $P_n$. The algebra $\mS_n$ is a noncommutative, non-Noetherian algebra of classical Krull dimension $2n$ and of global dimension $n$ which is not a domain. If the ground field $K$ has characteristic zero then the algebra $\mS_n$ is canonically isomorphic to the algebra $K< \frac{\der}{\der x_1}, ..., frac{\der}{\der x_n}, \int_1, ..., \int_n>$ of scalar integro-differential operato…
Let $R$ be the associative $k$-algebra generated by two elements $x$ and $y$ with defining relation $yx=1$. A complete description of simple modules over $R$ is obtained by using the results of Irving and Gerritzen. We examine the short exact sequence $0\rightarrow U\rightarrow E \rightarrow V\rightarrow 0$, where $U$ and $V$ are simple $R$-modules. It shows that nonsplit extension only occurs when both $U$ and $V$ are one-dimensional, or, under certain condition, $U$ is infinite-dimensional and $V$ is one-dimensional.
Standard noncommutative Gröbner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gröbner basis procedures for one-sided ideals in finitely presented noncommutative algebras over fields. The polynomials defining a $K$-algebra $A$ as a quotient of a free $K$-algebra are combined with the polynomials defining a one-sided ideal $I$ of $A$, by using a tagging notation. Standard noncommutative Gröbner basis techniques can then be applied to the mixed set of polynomials, thus calculating $A/I$ whilst working in a free structure, …
In this paper, we first prove that the absorption law for one-sided inverses along an element holds, deriving the absorption law for the inverse along an element. We then apply this result to obtain the absorption law for the inverse along different elements. Also, the reverse order law and the existence criterion for the inverse along an element are given by centralizers in a ring. Finally, we characterize the Moore-Penrose inverse by one-sided invertibilities in a ring with involution.
In this paper, we investigate some properties of annihilator $(b,c)$-inverses in an arbitrary ring. We demonstrate that one-sided annihilator $(b,c)$-inverses of elements in arbitrary rings may behave differently in contrast to one-sided $(b,c)$-inverses. Also, we discuss intertwining property, absorption law, reverse order law, and Cline's formula for annihilator $(b,c)$-inverses. As applications, we improve and extend some known results to $(b,c)$-inverses. In particular, we derive an equivalent condition of intertwining property for $(b,c)$-inverses in semigroups.
We give a thorough structural analysis of the principal one-sided ideals of arbitrary semigroups, and then apply this to full transformation semigroups and symmetric inverse monoids. One-sided ideals of these semigroups naturally occur as semigroups of transformations with restricted range or kernel.
In this paper we prove that one-sided Duo rings are (two-sided) McCoy. By doing so, we are then able to explicitly describe some of these ring element annihilators of polynomials in McCoy rings. We conclude the paper by showing the place of these results in the literature by way of an extension of a convenient diagram from a paper by Camillo and Nielsen.
In this paper, we derive some necessary and sufficient solvability conditions for some systems of one sided coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. We also give the expressions of the general solutions to these systems when they are solvable. Moreover, we provide some numerical examples to illustrate our results. The findings of this paper extend some known results in the literature.
In classical and real algebraic geometry there are several notions of the radical of an ideal I. There is the vanishing radical defined as the set of all real polynomials vanishing on the real zero set of I, and the real radical defined as the smallest real ideal containing I. By the real Nullstellensatz they coincide. This paper focuses on extensions of these to the free algebra R<x,x^*> of noncommutative real polynomials in x=(x_1,...,x_g) and x^*=(x_1^*,...,x_g^*). We work with a natural notion of the (noncommutative real) zero set V(I) of a left ideal I in the free algebra. The vanishing r…
This article is devoted to the noncommutative version of the Laplace transformation. New types of the direct and inverse transformations of the Laplace type over general Cayley-Dickson algebras, in particular, also the skew field of quaternions and the octonion algebra are investigated. Examples are given. Theorems are proved about properties of such transformations and also theorems about images and originals in conjunction with operations of multiplication, differentiation, integration, convolution, shift and homothety. The noncommutative analog of the Mellin transformation is studied as wel…
We consider the max-plus analogue of the eigenproblem for matrix pencils Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible values of lambda), which is a finite union of intervals, can be computed in pseudo-polynomial number of operations, by a (pseudo-polynomial) number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between Ax and lambda Bx. The spectrum is obtained as the zero level set of this function.
Let $n \ge 3$ be an integer. Let $P_n = \{1, 2, 3, ..., n-1, n \}$ and let $S_n$ be the symmetric group of permutations on $P_n$. Motivated by the theory of discrete dynamical systems on the interval, we associate each permutation $\si_n$ in $S_n$ a (zero-one) Petrie matrix $M_{\si_n,n-1}$ in $GL(n-1,{\mathbb{R}})$ (which is generally not the same as the usual permutation matrix). Then, for any two permutations $\si_n$ and $ρ_n$ in $S_n$, the notions of right, left and two-sided similarities (and weak similarities respectively) of $\si_n$ and $ρ_n$ are introduced using the similarities (and th…
Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the Prime Ideal Principle for commutative rings due to T.Y. Lam and the author. Old and new "maximal implies prime" results are presented, with results touching on annihilator ideals, polynomial identity rings, the Artin-Rees property, Dedekind-finite rings, principal ideals generated by normal elements, strongly noetherian algebras, and just infinite algebras.
In this paper, we first prove that if a is both left (b, c)-invertible and left (c, b)-invertible, then a is both (b, c)-invertible and (c, b)-invertible in a *-monoid, which generalized the recent result about the inverse along an element by Wang and Mosic, under the conditions (ab)* = ab and (ac)* = ac. In addition, we consider that ba is (c, b)- invertible, and at the same time ca is (b, c)-invertible under the same conditions, which extend the related results about Moore-Penrose inverses by Chen et al. to (b, c)-inverses. As applications, we obtain that under condition (a2)* = a2, a is an …
We study the complexity of computation in finitely generated free left, right and two-sided adequate semigroups and monoids. We present polynomial time (quadratic in the RAM model of computation) algorithms to solve the word problem and compute normal forms in each of these, and hence also to test whether any given identity holds in the classes of left, right and/or two-sided adequate semigroups.
