Conductor, projectivity and injectivity

All rings are assumed to be associative and with nonzero identity element, and all modules are assumed to be unitary. A ring is said to be right (resp., left) invariant if all right (resp., left) ideals of it are ideals. Expressions such as “an invariant ring” mean that the corresponding right and left conditions hold. A right module M over a ring A is called a multiplication module if for every submodule N of M , there exists an ideal B of the ring A such that N = MB. A module MA is said to be flat if for every left A-module L, the natural group homomorphism M ⊗A L is a monomorphism. If all submodules of any right or left module over a ring A are flat, then one says that the weak global dimension of the ring A does not exceed 1; in this case, we write w.gl.dim(A) ≤ 1 for brevity. In [7], it is proved that every faithful multiplication module over a commutative ring is a flat module. If A is a commutative ring, then w.gl.dim(A) ≤ 1 if and only if A is a distributive semiprime ring [5]. If A is an invariant semiprime ring, then w.gl.dim(A) ≤ 1 if and only if A is a distributive ring [13, Theorem 7.31]. The main result of the present paper is Theorem 1.

Totally acyclic complexes

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.

The purpose of this paper is to introduce two new classes of rings that is closely related to the classes of piecewise Noetherian domains and piecewise w-Noetherian domains. Let is a commutative ring and is a divided prime ideal of Let be a ring with total quotient ring T(R), and set such that for every and every Then is a ring homomorphism from T(R) into and restricted to R is also a ring homomorphism from R into given by for every At this setting, we introduce the notions of -piecewise Noetherian rings and -piecewise w-Noetherian rings and we show that the theories of -piecewise Noetherian (resp. -piecewise w-Noetherian) rings resemble those of piecewise Noetherian (resp. piecewise w-Noetherian) domains.

Let R be a commutative unital ring and E a unital R-module. Then the canonical injective ring homomorphism from R into the idealization R(+) E is a minimal ring homomorphism if and only if E is a simple R-module. For E nonzero, R(+)E is not (R-algebra isomorphic to) an overring of R. If E 1 and E 2 are nonisomorphic simple R-modules, then R(+) E 1 and R(+) E 2 give minimal ring extensions of R which are not isomorphic as R-algebras. The ring of dual numbers over R is a minimal ring extension of R ⇔ R × R is a minimal ring extension of R ⇔ R is a field.

This paper discusses the concept of an n-valued group. The theory that is developed here can be regarded as having more precise conditions than those in the much more developed theory of hypergroups; it is also more oriented towards algebraic and topological ideas. A number of methods for constructing examples are given and the classification of the algebraic 2-valued group structures on the Riemann sphere is presented. We introduce the (purely algebraic) notion of an n-Hopf algebra and show that the ring of functions on an n-valued group has an n-Hopf algebra structure. The crucial property of an n-Hopf algebra is that the diagonal is an n-ring homomorphism that has a weaker multiplicative property than that satisfied by a ring homomorphism. This concept arises in a number of other contexts, in particular as a trace. Convolution of two ring homomorphisms from an n-Hopf algebra to any commutative ring yields an n-ring homomorphism. There is a duality between symmetric products of topological spaces and the set of all n-ring homomorphisms from the ring of functions on the space to C. The development of the properties of n-ring homomorphisms involves a number of combinatorial identities related closely to the classical properties of symmetric polynomials.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Multiplicative closure operations on ring extensions

Let $R$ and $S$ be commutative rings with unity, $f\colon R\rightarrow S$ a ring homomorphism and $J$ an ideal of $S$. Then the subring $R\bowtie ^fJ:=\lbrace (a,f(a)+j)\mid a\in R$ and $j\in J\rbrace $ of $R\times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$. In this paper, we determine when $R\bowtie ^fJ$ is a (generalized) filter ring.

The classes of <TEX>$G_C$</TEX> homological modules over commutative ring, where C is a semidualizing module, extend Holm and J<TEX>${\varnothing}$</TEX>gensen's notions of C-Gorenstein homological modules to the non-Noetherian setting and generalize the classical classes of homological modules and the classes of Gorenstein homological modules within this setting. On the other hand, transfer of homological properties along ring homomorphisms is already a classical field of study. Motivated by the ideas mentioned above, in this article we will investigate the transfer properties of C and <TEX>$G_C$</TEX> homological dimension.

The property of a generating set of a polynomial ideal or of an ideal of a free associative algebra over a commutative ring to be its Gröbner basis is kept by a flat (but not any) extension of the ground ring. The converse is proved in the broader context of standard bases for filtered modules.

A covariant functor has been defined in [1] which assigns to a commutative ring R an abelian monoid m(R). The elements of m(R) are equivalence classes (called homotopy classes) of homomorphisms of progenerator modules over R. The monoid homomorphism m(σ): m(R) → m(S) associated with a homomor-phism of rings σ: R → S is studied. Both necessary condi­tions and sufficient conditions for m(σ)1 {1} = {1}, for m(σ) to be injective and for m(σ) to be surjective are presented.

All rings considered in this paper will be commutative and Noetherian, will have identities, and will be assumed to be non-trivial unless otherwise specified. The letter A will always denote such a ring. It is to be assumed that ring homomorphisms respect identity elements.

Let K be a commutative ring with identity element and let Λi, i = 1, 2 be two K-projective associative algebras with identity element such that the map k → k · l of K into Λi is a monomorphism of K onto a K-direct summand of Λi, i = 1, 2. Let Λ = Λ1 ⊗ Λ2 and A be a two sided (Λ1, Λ2)-bimodule.

Let [Formula: see text] be a commutative ring. If the nilpotent radical [Formula: see text] of [Formula: see text] is a divided prime ideal, then [Formula: see text] is called a [Formula: see text]-ring. In this paper, we first distinguish the classes of nonnil-coherent rings and [Formula: see text]-coherent rings introduced by Bacem and Ali [Nonnil-coherent rings, Beitr. Algebra Geom. 57(2) (2016) 297–305], and then characterize nonnil-coherent rings in terms of [Formula: see text]-flat modules, nonnil-injective modules and nonnil-FP-injective modules. A [Formula: see text]-ring [Formula: see text] is called a [Formula: see text]-IF ring if any nonnil-injective module is [Formula: see text]-flat. We obtain some module-theoretic characterizations of [Formula: see text]-IF rings. Two examples are given to distinguish [Formula: see text]-IF rings and IF [Formula: see text]-rings.