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Let M be a finitely generated module on a local ring R and $\mathcal{F}: M_0\subset M_1\subset\cdots\subset M_t=M$ a filtration of submodules of M such that do < d1 < ⋯ < dt = d, where di = dim Mi. This paper is concerned with a non-negative integer $p_\mathcal{F}(M)$ which is defined as the least degree of all polynomials in n1, ..., nd bounding above the function \begin{eqnarray*} \ell(M/(x_1^{n_1}, \ldots, x_d^{n_d})M)-\sum_{i=0}^tn_1\cdots n_{d_i}e(x_1,\ldots, x_{d_i};M_i). \end{eqnarray*} We prove that $p_\mathcal F(M)$ is independent of the choice of good systems of parameters $\underline x=x_1, \ldots, x_d$. When $\mathcal{F}$ is the dimension filtration of M, we can use the polynomial type of Mi/Mi-1 and the dimension of the non-sequentially Cohen-Macaulay locus of M to compute $p_\mathcal{F}(M)$, and also to study the behavior of it under local flat homomorphisms. [ABSTRACT FROM AUTHOR]

A right R-module MR is called small injective if every homomorphism from a small right ideal to MR can be extended to an R-homomorphism from RR to MR. A ring R is called right small injective, if the right R-module RR is small injective. We prove that R is semiprimitive if and only if every simple right (or left) R-module is small injective. Further we show that the Jacobson radical J of a ring R is a noetherian right R-module if and only if, for every small injective module ER, E(ℕ) is small injective. [ABSTRACT FROM AUTHOR]

We introduce the concepts of τ-Max modules and left τ-Max rings which are torsion-theoretic analogues of Max modules and left Max rings. A generalization is obtained of an important theorem by Shock and used to characterize τ-Noetherian rings using τ-Max modules. We then characterize left τ-Max rings and obtain a torsion-theoretic version of a result by Hirano. We conclude with results on τ-short modules, introduced as a torsion analogue of a concept recently defined by Bilhan and Smith. [ABSTRACT FROM AUTHOR]

In this paper we describe the structure of prime and semiprime R-modules M such that R/AnnR(M) is artinian. The obtained results are then applied to describe the structure of prime and semiprime right comodules over a coring C under some assumption, where a right comodule M is said to be prime (semiprime) if the corresponding left *C-module is prime (semiprime). Finally we apply the results to coalgebras over commutative rings. [ABSTRACT FROM AUTHOR]