A tag module is a generalization, in any abelian category, of a torsion abelian group. The theory of such modules is developed, it is shown that countably generated tag modules are simply presented, and that Ulm's theorem holds for simply presented tag modules. Zippin's theorem is stated and proved for countably generated tag modules. 1. TAG-modules In the theory of torsion abelian groups, a di...

12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x1, . . . , xn is any maximal set of linearly independent elements of M . Prove that N is isomorphic to Rn and that the quotient M/N is a torsion Rmodule. (b) Prove conversely that if M contains a submodule N that is free of rank n (i.e., N › Rn) such that the quotient M/N is a torsion R-module then M ha...

Every vector space over a field K that has a finite spanning set has a finite basis: it is isomorphic to Kn for some n > 0. When we replace the scalar field K with a commutative ring A, it is no longer true that every A-module with a finite generating set has a basis: not all modules have bases. But when A is a PID, we get something nearly as good as that: (1) Every submodule of An has a basis ...

In this paper we introduce a generalization of M-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . We will use the structure of the radical of a module in  and get some suitable results about this class of modules. Also the relation between injective hull in  and this kind of modules will be investigated in this article.   For a module  we show...

in this paper we introduce a generalization of m-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . we will use the structure of the radical of a module in  and get some suitable results about this class of modules. also the relation between injective hull in  and this kind of modules will be investigated in this article.   for a module  we show...

We investigate the maximal finite length submodule of Breuil–Kisin prismatic cohomology a smooth proper formal scheme over $p$ -adic ring integers. This governs pathology phenomena in integral theories. Geometric applications include control, low degrees and mild ramifications, (1) discrepancy between two naturally associated Albanese varieties characteristic , (2) kernel specialization map éta...

the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d'subseteq m$ such that $n+d'$ is a strongly dense submodule of $m$...

It is proved that if R is a valuation domain with maximal ideal P and if RL is countably generated for each prime ideal L, then R R is separable if and only RJ is maximal, where J = ∩n∈NP . When R is a valuation domain satisfying one of the following two conditions: (1) R is almost maximal and its quotient field Q is countably generated (2) R is archimedean Franzen proved in [2] that R is separ...

All rings are commutative with 1 6= 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a, b ∈ R and m ∈M and abm ∈ N , then am ∈M -rad(N) or bm ∈M -rad(N) or ...

It is conjectured that for fixed A, r ≥ 1, and d ≥ 1, there is a uniform bound on the size of the torsion submodule of a Drinfeld A-module of rank r over a degree d extension L of the fraction field K of A. We verify the conjecture for r = 1, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of K. Moreover, we show that within an L-iso...