In ``New Proofs of the structure theorems for Witt Rings'', Lewis shows how the standard ring-theoretic results on the Witt ring can be deduced in a quick and elementary way from the fact that the Witt ring of a field is integral and from the specific nature of the explicit annihilating polynomials he provides. We will show in the present article that the same structure results hold for larger classes of commutative rings and not only for Witt rings. We will construct annihilating polynomials for these rings.
Let $R$ be a commutative and unital semi-local ring in which 2 is invertible. In this note, we show that anisotropic quadratic spaces over $R$ remain anisotropic after base change to any odd-degree finite étale extension of $R$. This generalization of the classical Artin-Springer theorem (concerning the situation where $R$ is a field) was previously established in the case where all residue fields of $R$ are infinite by I. Panin and U. Rehmann. The more general result presented here permits to extend a fundamental isotropy criterion of I. Panin and K. Pimenov for quadratic spaces over regular …
We expand on two existing characterizations of rings of Gorenstein (weak) global dimension zero and give two new characterizations of rings of finite Gorenstein (weak) global dimension. We also include the answer to a question of Y.~Xiang on Gorenstein weak global dimension of group rings.
We show a general decomposition theorem in Baer *-rings. As a consequence the vast majority of decompositions known in the algebra of bounded Hilbert space operators are generalized to Baer *-rings. There are also results which are new in the algebra of bounded Hilbert space operators. The model of summands in Wold-Słociński decomposition in Baer *-rings is given.
In Hai-Thin (2009), there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring $D$ is central (see Hai-Thin (2009, Th. 2.2)). Unfortunately, there is some mistake in the proof of this theorem. In this note we give the another proof of this theorem.
A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary semilocal ring, let S be a finite R-algebra of constant odd degree, which is {é}tale or generated by one element, and let q be a nonsingular R-quadratic form whose base ring extension q S is isotropic. We show that then q is already isotropic.
The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring
Mindanao State University-Iligan Institute of Technology（棉兰老州立大学伊利甘理工学院）；Blekinge Institute of Technology（布莱金厄理工学院）
We provide a variant of Baer's theorem about isomorphism of endomorphism rings of vector spaces over division rings, where the full endomorphism rings are replaced by some subrings of finitary maps.
We apply recent results on the rank of elements of rings to study the structure of generalized corner rings $aRa$, where $R$ is a unital ring and $a$ an element of $R$. We give a complete description of the structure of $aRa$ when $a^2$ has finite rank and provide an example to show that this assumption is necessary and optimal.
Let R be a unit-regular ring, and let a,b,c in R satisfy aba=aca. If ac and ba are group invertible, we prove that ac is similar to ba. Furthermore, if ac and ba are Drazin invertible, then their Drazin inverses are similar. For any n\times n complex matrices A,B,C with ABA = ACA ,we prove that AC and BA are similar if and only if their k-powers have the same rank. These generalize the known Flanders' theorem proved by Hartwig.
The classical theorem of Milnor on pullback rings states that the category of projective modules over a pullback ring is equivalent to a certain category of gluing triples consisting of projective modules. We prove an analogous result on the level of derived categories, where the equivalence has to be replaced by an epivalence.
We consider applications of a finitary version of the Affine Representability theorem, which follows from recent work of Belov-Kanel, Rowen, and Vishne. Using this result we are able to show that when given a finite set of polynomial identities, there is an algorithm that terminates after a finite number of steps which decides whether these identities force a ring to be commutative. We then revisit old commutativity theorems of Jacobson and Herstein in light of this algorithm and obtain general results in this vein. In addition, we completely characterize the homogeneous multilinear identities…
Congruence lattices of semiprime algebras from semi--degenerate congruence--modular varieties fulfill the equivalences from B. A. Davey`s well--known characterization theorem for $m$--Stone bounded distributive lattices, moreover, changing the cardinalities in those equivalent conditions does not change their validity. I prove this by transferring Davey`s Theorem from bounded distributive lattices to such congruence lattices through a certain lattice morphism and using the fact that the codomain of that morphism is a frame. Furthermore, these equivalent conditions are preserved by finite direc…
We present two criteria for a group $G$ to satisfy the following statements: any $G$-graded gr-prime (gr-semiprime) right gr-Goldie ring admits a gr-semisimple graded right classical quotient ring. The criterion for gr-semiprime rings is that the group $G$ is periodic. Actually, the sufficiency of periodicity was proved by the author in 2011 and the necessity of it follows from the well-known counterexample (1979). The main result of the paper concerns the gr-prime case. In this case, Goodearl and Stafford proved the graded version of Goldie's Theorem for rings graded by an Abelian group (2000…
A proof of a theorem of M. Hertweck presented during a seminar in January 2013 in Stuttgart is given. The proof is based on a preprint given to me by Hertweck. Let $R$ be a commutative ring, $G$ a finite group, $N$ a normal $p$-subgroup of $G$ and denote by $RG$ the group ring of $G$ over $R$. It is shown that a torsion unit $u$ in $\mathbb{Z}G$ mapping to the identity under the natural homomorphism $\mathbb{Z}G \rightarrow \mathbb{Z}G/N$ is conjugate in the unit group of $\mathbb{Z}_pG$ to an element in $N$. Here $\mathbb{Z}_p$ denotes the $p$-adic integers. The result is achieved proving a r…
Let $A_1,\ldots,A_s$ be unitary commutative rings which do not have non-trivial idempotents and let $A=A_1\oplus\cdots\oplus A_s$ be their direct sum. We describe all idempotents in the $2\times 2$ matrix ring $M_2(A[[X]])$ over the ring $A[[X]]$ of formal power series with coefficients in $A$ and in arbitrary set of variables $X$. We apply this result to the matrix ring $M_2({\mathbb Z}_n[[X]])$ over the ring ${\mathbb Z}_n[[X]]$ for an arbitrary positive integer $n$ greater than 1. Our proof is elementary and uses only the Cayley-Hamilton theorem (for $2\times 2$ matrices only) and, in the s…
A well-known theorem by S.A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x; D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x; D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p>0.
We give a sufficient condition for a bi-invariant weight on a Frobenius bimodule to satisfy the extension property. This condition applies to bi-invariant weights on a finite Frobenius ring as a special case. The complex-valued functions on a Frobenius bimodule are viewed as a module over the semigroup ring of the multiplicative semigroup of the coefficient ring.
In this paper we study non-central almost subnormal subgroups of the multiplicative group of a division ring satisfying a non-zero generalized rational identity. The main result generalizes Chiba's theorem on subnormal subgroups. As an application, we get a theorem on almost subnormal subgroups satisfying a generalized algebraic rational identity. The last theorem has several corollaries which generalize completely or partially some previous results.
We compute the $k$th power-sums (for all $k>0$) over an arbitrary finite unital ring $R$. This unifies and extends the work of Brawley, Carlitz, and Levine for matrix rings [Duke Math. J. 1974], with folklore results for finite fields and finite cyclic groups, and more general recent results of Grau and Oller-Marcen for commutative rings [Finite Fields Appl. 2017]. As an application, we resolve a conjecture by Fortuny Ayuso, Grau, Oller-Marcen, and Rua on zeta values for matrix rings over finite commutative rings [Internat. J. Algebra Comput. 2017]. We further recast our main result via zeta v…
Rings with Clean-Like Properties: Endomorphism, Matrix and Structural Theorems
Kazan (Volga Region) Federal University（喀山（伏尔加沿岸）联邦大学）；Faculty of Mathematics and Mechanics, Tomsk State University（托木斯克国立大学数学与力学系）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）
We study a topological generalization of ideal co-maximality in topological rings and present some of its properties, including a generalization of the Chinese remainder theorem. Using the hyperspace uniformity, we prove a stronger version of this theorem concerning infinitely many ideals in supercomplete, pseudo-valuated rings. Finally we prove two interpolation theorems.
Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient i…
