We initiate the study of 1-torsion of finite modules over two-sided noetherian semiperfect rings. In particular, we give a criterion for determining when the 1-torsion submodule contains minimal generators of the module. We also provide an explicit construction for a projective cover of the submodule generated by the torsion elements in the top of the module. Some of the obtained results hold w...

Let R be a commutative Noetherian ring of dimension d and M cancellative torsion-free seminormal monoid. Then: (1) A type R[d,m,n] P projective A[M]-module rank r≥max {2,d+1}. Then the action E(A[M]⊕P) on (A[M]⊕P) is transitive (2) Assume (R,m,K) regular local containing field k such that either char k=0 or k=p tr-deg K∕

Let R be a commutative noetherian ring. We study the behavior of injectiveand at dimension of R-modules under the functors HomR(-,-) and -×R-.

Several of the fundamental theorems about algebraic K, and Kr are concerned with finding unimodular elements, that is, elements of a projective module which generate a free summand. In this paper we use the notion of a basic element (in the terminology of Swan [22]) to extend these theorems to the context of finitely generated modules. Our techniques allow a simplification and strengthening of ...

This a first step to develop a theory of smooth, étale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of pseudo finite type. Among our results we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitab...

let r be a commutative noetherian ring. we study the behavior of injectiveand at dimension of r-modules under the functors homr(-,-) and -×r-.

In this paper, we study some ring theoretic properties of the amalgamated duplication ring $Rbowtie I$ of a commutative Noetherian ring $R$ along an ideal $I$ of $R$ which was introduced by D'Anna and Fontana. Indeed, it is determined that when $Rbowtie I$ satisfies Serre's conditions $(R_n)$ and $(S_n)$, and when is a normal ring, a generalized Cohen-Macaulay ring and finally a filter ring.

Let $R$ be a ring, $sigma$ be an endomorphism of $R$ and $M_R$ be a $sigma$-rigid module. A module $M_R$ is called quasi-Baer if the right annihilator of a principal submodule of $R$ is generated by an idempotent. It is shown that an $R$-module $M_R$ is a quasi-Baer module if and only if $M[[x]]$ is a quasi-Baer module over the skew power series ring $R[[x,sigma]]$.

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...

In this paper certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring R is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of RR into R (N) R can be extended to RR. Also, known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is...