Let I denote an ideal of a local Gorenstein ring (R, m). Then we show that the local cohomology module H I (R), c = height I, is indecomposable if and only if V (Id) is connected in codimension one. Here Id denotes the intersection of the highest dimensional primary components of I. This is a partial extension of a result shown by Hochster and Huneke in the case I the maximal ideal. Moreover th...

Let R be a commutative ring with identity. It often happens that M1 ⊕ · · · ⊕ Ms ∼= N1 ⊕ · · · ⊕ Nt for indecomposable R-modules M1, . . . , Ms and N1, . . . , Nt with s 6= t. This behavior can be captured by studying the commutative monoid {[M ] |M is an R-module} of isomorphism classes of R-modules with operation given by [M ]+[N ] = [M⊕N ]. In this mostly self-contained exposition, we introd...

Finitely generated modules with finite F -representation type (or FFRT for short) over Noetherian (local) rings of prime characteristic p are studied. If a ring R has FFRT or, more generally, if a faithful R-module has FFRT, then tight closure commutes with localizations over R. We also define F -contributors and use them to give an effective way to characterize tight closure. Then we show lime...

Let = ZG be the integer group ring of a group, G, of prime order. A main result of this note is that every -module with a free underlying abelian group decomposes into a direct sum of copies of the well-known indecomposable -lattices of finite rank. The first part of the proof reduces the problem to one about countably generated modules, and works in a wider context of suitably restricted modul...

Let G be the Klein four-group and let k an arbitrary field of characteristic 2. A classification indecomposable kG-modules is known. We calculate relative cohomology groups Hχi(G,N) for every kG-module N, where χ set proper subgroups in G. This extends work Pamuk Yalcin to with non-trivial coefficients. also show that all cup products strictly positive degree Hχ*(G,k) are trivial.

An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, finite direct sum decomposition consisting of objects with local endomorphism rings—is known as Krull-Schmidt category. A Hom-finite is an for there commutative unital ring k, such that Hom-set length k-module. The aim this note to provide proof Krull-Schmidt, if and only it split idempotents, indecomposa...

If A is a primitive matrix, then there is a smallest power of A (its fully indecomposable exponent) that is fully indecomposable, and a smallest power of A (its strict fully indecomposable exponent) starting from which all powers are fully indecomposable. We obtain bounds on these two exponents.

Abstract We present a sufficient condition for the $kG$ -Scott module with vertex $P$ to remain indecomposable under Brauer construction any subgroup $Q$ of as $k[Q\,C_G(Q)]$ -module, where $k$ is field characteristic $2$ , and semidihedral -subgroup finite group $G$ . This generalizes results cases abelian or dihedral. The indecomposability defined by R. Kessar, N. Kunugi Mitsuhashi. motivatio...

In this handout we will briefly explore the topic of projective modules in a bit more detail than we covered in class. Throughout R is a commutative ring. Recall that, by definition, a projective module is an R-module that is a direct summand of a free R-module. As mentioned in class, if the ring R is decomposable, e.g., R = R1 ⊕R2 is a direct sum of rings, then there are many examples of non-f...