Introduction 5 0.1. What is Commutative Algebra? 5 0.2. Why study Commutative Algebra? 5 0.3. Acknowledgments 7 1. Commutative rings 7 1.1. Fixing terminology 7 1.2. Adjoining elements 10 1.3. Ideals and quotient rings 11 1.4. The monoid of ideals of R 14 1.5. Pushing and pulling ideals 15 1.6. Maximal and prime ideals 16 1.7. Products of rings 17 1.8. A cheatsheet 19 2. Galois Connections 20 2...

In our recent work we gave a treatment of certain aspects of multiplication modules, projective modules, flat modules and cancellation-like modules via idealization. The purpose of this work is to continue our study and develop the tool of idealization, particularly in the context of closed, divisible injective, and simple modules. We determine when a ring R (M), the idealization of M , is a qu...

Let p be an odd prime, and let OK be the ring of integers in a finite extension K/Qp. Breuil has classified finite flat group schemes of type (p, . . . , p) over OK in terms of linear-algebraic objects that have come to be known as Breuil modules. This classification can be extended to the case of finite flat vector space schemes G over OK . When G has rank one, the generic fiber of G correspon...

In this paper, we show the existence of copure injective preenvelopes over noetherian rings and copure flat preenvelopes over commutative artinian rings. We use this to characterize n-Gorenstein rings. As a consequence, if the full subcategory of strongly copure injective (respectively flat) modules over a left and right noetherian ring R has cokernels (respectively kernels), then R is 2-Gorens...

let $r$ be a commutative noetherian ring with non-zero identity, $fa$ an ideal of $r$, and $x$ an $r$--module. here, for fixed integers $s, t$ and a finite $fa$--torsion $r$--module $n$, we first study the membership of $ext^{s+t}_{r}(n, x)$ and $ext^{s}_{r}(n, h^{t}_{fa}(x))$ in the serre subcategories of the category of $r$--modules. then, we present some conditions which ensure the exi...

we introduce the notions of t-dual rickart and strongly t-dual rickart modules. we provide several characterizations and investigate properties of each of these concepts. it is shown that every free (resp. finitely generated free) $r$-module is t-dual rickart if and only if $overline{z}^2(r)$ is a  direct summand of $r$ and end$(overline{z}^2(r))$ is a semisimple (resp. regular) ring. it is sho...