Applications of reduced and coreduced modules II

This is the second in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We give the necessary and sufficient conditions in terms of $I$-reduced and $I$-coreduced $R$-modules for the functors $\text{Hom}_R(R/I, -)$ and $\Gamma_I$, the $I$-torsion functor, on the abelian full subcategories of the category of all $R$-modules to be radicals. These conditions: 1) subsume and unify many results which were proved on a case-by-case basis, 2) provide a setting for the generalisation of Jans' correspondence of an idempotent ideal of a ring with a torsion-torsionfree class, 3) provide answers to open questions that were posed by Rohrer, and 4) lead to a new radical class of rings.Comment: 20 page

A contribution to the theory of prime modules

This thesis is aimed at generalizing notions of rings to modules. In par-ticular, notions of completely prime ideals, s-prime ideals, 2-primal rings and nilpotency of elements of rings are respectively generalized to completely prime submodules and classical completely prime submodules, s-prime submodules, 2-primal modules and nilpotency of elements of modules. Properties and rad-icals that arise from each of these notions are studied

Reduced submodules of finite dimensional polynomial modules

Let $k$ be a field with characteristic zero, $R$ be the ring $k[x_1, \cdots, x_n]$ and $I$ be a monomial ideal of $R$. We study the Artinian local algebra $R/I$ when considered as an $R$-module $M$. We show that the largest reduced submodule of $M$ coincides with both the socle of $M$ and the $k$-submodule of $M$ generated by all outside corner elements of the Young diagram associated with $M$. Interpretations of different reduced modules is given in terms of Macaulay inverse systems. It is further shown that these reduced submodules are examples of modules in a torsion-torsionfree class, together with their duals; coreduced modules, exhibit symmetries in regard to Matlis duality and torsion theories. Lastly, we show that any $R$-module $M$ of the kind described here satisfies the radical formula.Comment: 19 page

Weakly-morphic modules

Let $R$ be a commutative ring, $M$ an $R$-module and $\varphi_a$ be the endomorphism of $M$ given by right multiplication by $a\in R$. We say that $M$ is {\it weakly-morphic} if $M/\varphi_a(M)\cong \ker(\varphi_a)$ as $R$-modules for every $a$. We study these modules and use them to characterise the rings $R/\text{Ann}_R(M)$, where $\text{Ann}_R(M)$ is the right annihilator of $M$. A kernel-direct or image-direct module $M$ is weakly-morphic if and only if each element of $R/\text{Ann}_R(M)$ is regular as an endomorphism element of $M$. If $M$ is a weakly-morphic module over an integral domain $R$, then $M$ is torsion-free if and only if it is divisible if and only if $R/\text{Ann}_R(M)$ is a field. A finitely generated $\Bbb Z$-module is weakly-morphic if and only if it is finite; and it is morphic if and only if it is weakly-morphic and each of its primary components is of the form $(\Bbb Z_{p^k})^n$ for some non-negative integers $n$ and $k$.Comment: 20 page