Semilocal Ring Research Articles

Gersten conjecture for equivariant K-theory and applications

For a reductive group scheme G over a regular semi-local ring A, we prove the Gersten conjecture for the equivariant K-theory. As a consequence, we show that if F is the field of fractions of A, then \({K^G_0(A) \cong K^G_0(F)}\), generalizing the analogous result for a dvr by Serre (Inst Hautes Études Sci Publ Math 34:37–52, 1968). We also show the rigidity for the K-theory with finite coefficients of a Henselian local ring in the equivariant setting. We use this rigidity theorem to compute the equivariant K-theory of algebraically closed fields.

Self-Orthogonal Modules of Finite Projective Dimension

Let R be a ring and R ω a self-orthogonal module. We introduce the notion of the right orthogonal dimension (relative to R ω) of modules. We give a criterion for computing this relative right orthogonal dimension of modules. For a left coherent and semilocal ring R and a finitely presented self-orthogonal module R ω, we show that the projective dimension of R ω and the right orthogonal dimension (relative to R ω) of R/J are identical, where J is the Jacobson radical of R. As a consequence, we get that R ω has finite projective dimension if and only if every left (finitely presented) R-module has finite right orthogonal dimension (relative to R ω). If ω is a tilting module, we then prove that a left R-module has finite right orthogonal dimension (relative to R ω) if and only if it has a special ω⊥-preenvelope.

SK1 of Azumaya algebras over Hensel Pairs

Let A be an Azumaya algebra of constant rank n 2 over a Hensel pair (R, I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK1(A) coincides with its reduction SK1(A/I A). This generalizes a result of Hazrat (J Algebra 305:687–703, 2006) to non-local Henselian rings.

On semilocal rings

In this paper, semilocal rings are characterized in different ways; in particular, it is proved that a ring R is semilocal if and only if every descending chain of principal right ideals of R, a 0 R ⊇ a 1 R ⊇ a 2 R D... ⊇ a n R ⊇ with a i+1 = a i - a i b i a i eventually terminates. Then modules with semilocal endomorphism rings are characterized by chain conditions.

A Gersten-Witt complex for hermitian Witt groups of coherent algebras over schemes II: Involution of the second kind

AbstractLet X be a regular noetherian scheme of finite Krull dimension with involution σ and an Azumaya algebra over X with involution τ of the second kind with respect to σ. We construct a hermitian and a skew-hermitian Gersten-Witt complex for (, τ) and show that these complexes are exact if X = Spec R is the spectrum of a regular semilocal ring R of geometric type, such that R is a quadratic étale extension of the fixed ring of σ.

N-adic p-basis and Regular semi-local ring

Let (R,m) be a regular local ring of prime characteristic p and R' a Noetherian subring of R such that Rp⊂R'. In the previous paper Furuya and Niitsuma [1], introducing the concept of m-adic p-basis, the first and the second authors proved the following theorem: The regular local ring R has an m-adic p-basis over the subring R' if and only if R' is a regular local ring. In this paper, we generalize this result to semi-local rings.

ON SB-RINGS

In this paper, we introduce a new class of rings, SB-rings. We establish various properties of this concept. These shows that, in several respects, SB-rings behave like rings satisfying unit 1-stable range. We will give necessary and sufficient conditions under which a semilocal ring is a SB-ring. Furthermore, we extend these results to exchange rings with all primitive factors artinian. For such rings, we observe that the concept of the SB-ring coincides with Goodearl-Menal condition. These also generalize the results of Huh et al., Yu and the author on rings generated by their units.

On the finiteness properties of Matlis duals of local cohomology modules

Let R be a complete semi-local ring with respect to the topology defined by its Jacobson radical, a an ideal of R, and M a finitely generated R-module. Let D R (−) := Hom R (−, E), where E is the injective hull of the direct sum of all simple R-modules. If n is a positive integer such that Ext R j (R/a, D R (H a t (M))) is finitely generated for all t > n and all j ⩾ 0, then we show that Hom R (R/a, D R (H a n (M))) is also finitely generated. Specially, the set of prime ideals in Coass R (H a n (M)) which contains a is finite.

Finitely Generated Modules over Semilocal Rings and Characterizations of (Semi-)Perfect Rings

Lomp (9) has studied finitely generated projective modules over semilocal rings. He obtained the following: finitely generated projective modules over semilocal rings are semilocal. We shall give necessary and sucient conditions for finitely generated modules to be semilocal modules. By using a lifting property, we also give characterizations of right perfect (semiperfect) rings. Our main results can be summarized as follows: (1) Let M be a finitely generated module. Then M has finite hollow dimension if and only if M is weakly supplemented if and only if M is semilocal. (2) A ring R is right perfect if and only if every flat right R-module is lifting and every right R-module has a flat cover if and only if every quasi-projective right R-module is lifting. (3) A ring R is semiperfect if and only if every finitely generated flat right R-module is lifting if and only if RR satisfies the lifting property for simple factor modules. In this note, all rings R considered are associative rings with identity and all modules are unital right R-modules unless indicated otherwise. For a module M, Rad(M), Soc(M), E(M), EndR(M) are the (Jacobson) radical, socle, injective hull and endomorphism ring of M, respectively. Let M be a module and let K be a submodule of M. K is called small submodule (or superfluous submodule) of M, abbreviated K M, if, for every submodule L M, K + L = M implies L = M. Let N1 N2 M. N1 is a co-essential submodule of N2 in M, abbreviated N1 c N2 in M, if N2/N1 M/N1. A submodule N of M is said to be co-closed in M (or a co-closed submodule of M), if N has no proper co-essential submodule in M. i.e., N 0 c N in M implies N = N 0 . Let N1 N2 M. N1 is said to be a co-closure of N2 in M if N1 is a co-closed submodule of M with N1 c N2 in M. Any submodule of a module has a closure. However, a co-closure does not exist in general, for example, 2Z does not have a co-closure in ZZ. Let M be a module and let N and L be submodules of M. N is called a supplement of L if M = N + L and N L N. Note that any supplement

THE STRUCTURE OF SEMIPERFECT RINGS

Let R be a ring with identity <TEX>$1_R$</TEX> and let U(R) denote the group of all units of R. A ring R is called locally finite if every finite subset in it generates a finite semi group multiplicatively. In this paper, some results are obtained as follows: (1) for any semilocal (hence semiperfect) ring R, U(R) is a finite (resp. locally finite) group if and only if R is a finite (resp. locally finite) ring; U(R) is a locally finite group if and only if U<TEX>$(M_n(R))$</TEX> is a locally finite group where <TEX>$M_n(R)$</TEX> is the full matrix ring of <TEX>$n{\times}n$</TEX> matrices over R for any positive integer n; in addition, if <TEX>$2=1_R+1_R$</TEX> is a unit in R, then U(R) is an abelian group if and only if R is a commutative ring; (2) for any semiperfect ring R, if E(R), the set of all idempotents in R, is commuting, then <TEX>$R/J\cong\oplus_{i=1}^mD_i$</TEX> where each <TEX>$D_i$</TEX> is a division ring for some positive integer m and |E(R)|=<TEX>$2^m$</TEX>; in addition, if 2=<TEX>$1_R+1_R$</TEX> is a unit in R, then every idempotent is central.

Reduction of structure for torsors over semilocal rings

Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is semisimple or k is normal and noetherian. We show that G has a finite k-subgroup S such that the natural map H 1(R, S) → H 1(R, G) is surjective for every semilocal ring R containing k. In other words, G-torsors over Spec(R) admit reduction of structure to S. We also show that the natural map H 1(X, S) → H 1(X, G) is surjective in several other contexts, under suitable assumptions on the base ring k, the scheme X/k and the group scheme G/k. These results have already been used to study loop algebras and essential dimension of connected algebraic groups in prime characteristic. Additional applications are presented at the end of this paper.

Subgroups of SLn over a semilocal ring

In the present paper, it is proved that if R is a commutative semilocal ring all the residue fields of which contain at least 3n + 2 elements, then for every subgroup H of the special linear group SL(n, R), n ≥ 3, containing the diagonal subgroup SD(n, R) there exists a unique D-net σ of ideals of R such that Γ(σ)≤H≤NΓ(σ). In works by Z. I. Borewicz and the author, similar results were established for GL n over semilocal rings and for SL n over fields. Later I. Hamdan obtained a similar description for the very special case of uniserial rings. Bibliography: 76 titles.

Witt equivalence of semilocal Dedekind domains in global fields

We examine a condition for two semilocal Dedekind rings, the fields of fractions of which are global fields, to be Witt equivalent. To solve the problem we generalize the notion of a Hilbert-symbol equivalence introduced in [11] and prove that a Witt equivalence is equivalent to a Hilbert-symbol equivalence. As a result we describe a Witt equivalence in terms of field invariants.

Indecomposable flat cotorsion modules

An additive functor from the category of flat right R-modules to the category of abelian groups is continuous if it is isomorphic to a functor of the form−⊗R M, where M is a left R-module. It is shown that for any simple subfunctor A of−⊗ M there is a unique indecomposable flat cotorsion module UR for which A(U)≠0. It is also proved that every subfunctor of a continuous functor contains a simple subfunctor. This implies that every flat right R-module may be purely embedded into a product of indecomposable flat cotorsion modules. If CE(R) is the cotorsion envelope of RR and S= End;R CE(R), then a local ring monomorphism is constructed from R/J(R) to S/J(S). This local morphism of rings is used to associate a semiperfect ring to any semilocal ring. It also proved that if R is a semilocal ring and M a simple left R-module, then the functor−⊗R M on the category of flat right R-modules is uniform, and therefore contains a unique simple subfunctor.

CLASSIFYING PROJECTIVE MODULES OVER SOME SEMILOCAL RINGS

We investigate the module theory of a certain class of semilocal rings connected with nearly simple uniserial domains. For instance, we classify finitely presented and pure-projective modules over these rings and calculate their projective dimension.

Generatrices and relations of a generalized orthogonal group over commutative semilocal rings without identity

The description of orthogonal groups (and close to them ones) in terms of generatrices and relations represents one of the main problems of the combinatorial theory of linear groups. This paper is dedicated to the mentioned question; namely, here we find generating elements and defining relations of a generalized orthogonal group O◦(n,R), n ≥ 2, over a commutative semilocal ring R (in general, without identity) under certain natural constraints. In order to formulate the problem exactly, let us define several necessary notions. LetΛ be an arbitrary associative ring and let ◦ be its quasi-multiplication (i. e., x ◦ y = x+ xy + y). An element α from Λ is said to be quasi-invertible, if with certain α′ ∈ Λ, α ◦ α′ = 0 = α′ ◦ α. If α is quasi-invertible, then one can uniquely determine its quasi-inverse α′. The set of all quasi-invertible elements Λ◦ from Λ forms a group with respect to the composition ◦ (where the identity is zero). We consider the case when Λ = M(n,R) is a complete matrix ring over a ring T which is (associative-)commutative and not necessarily with identity. Let mean the transposition in Λ. The set of quasi-invertible matrices from Λ such that a′ = a forms a group with respect to the matrix quasi-multiplication. We denote it by O◦(n,R) and call it a generalized orthogonal group of the degree n over the ring R. Note that if R contains the identity, then the mapping O(n,R) → O◦(n,R), E + a → a, where E is the unit matrix of the order n, defines an isomorphism. So the introduced groupO◦(n,R) generalizes the notion of a usual orthogonal group to the most general cases of associative-commutative rings R. Let now R be a commutative semilocal ring (not necessarily with identity) and let J = J(R) be its Jacobson radical. By definition it means that R/J ∼= k1 ⊕ · · · ⊕ km, where ki are certain fields (i = 1, . . . ,m). Let Ri stand for the complete preimage of the addend ki under the natural epimorphism R → R = R/J, x → x = x+ J. (1)

Comaximal graph of commutative rings

Let R be a commutative ring with identity. Let Γ ( R ) be a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if R a + R b = R . In this paper we consider a subgraph Γ 2 ( R ) of Γ ( R ) which consists of non-unit elements. We look at the connectedness and the diameter of this graph. We completely characterize the diameter of the graph Γ 2 ( R ) ∖ J ( R ) . In addition, it is shown that for two finite semi-local rings R and S, if R is reduced, then Γ ( R ) ≅ Γ ( S ) if and only if R ≅ S .

Projective modules are determined by their radical factors

We show that two infinitely generated projective modules are isomorphic whenever they have isomorphic factors modulo their Jacobson radical. Some applications of the result to semilocal rings with indecomposable non-finitely generated projective modules are given.

Semilocal rings with Engel conditions

The relation between the Engel structure of a semilocal ring and that of its multiplicative group is investigated. Suppose that every local ring whose multiplicative group satisfies an m-Engel condition for some positive integer m is an f (m)-Engel ring for some function f . It is proved that under this condition a corresponding statement holds for every semilocal ring which is generated by its multiplicative group.

Local Morphisms and Modules with a Semilocal Endomorphism Ring

We study local morphisms in the setting of general noncommutative rings. In particular, we apply local morphisms to study endomorphism rings of modules. We use our constructions to determine classes of modules with semilocal endomorphism rings. For instance, we prove that every finitely presented right module over a semilocal ring has a semilocal endomorphism ring.