The purpose of this study was to compile and statistically analyze the results of research
studies that examined students' problem-solving skills in worksheets. The research method …

Consider a non-trivial fiber product $ R= S\times _kT $ of local rings $ S $, $ T $ with
common residue field $ k $. Given two finitely generated $ R $-modules $ M $ and $ N $, we …

Motivated by a result of Araya, we extend the Auslander–Reiten duality theorem to Cohen–
Macaulay local rings. We also study the Auslander–Reiten conjecture, which is rooted in …

We introduce the notion of Burch submodules and weakly m-full submodules of modules
over a local ring (R, m) and study their properties. One of our main results shows that Burch …

Auslander's depth formula for pairs of Tor-independent modules over a regular local ring,
depth (M⊗ RN)= depth M+ depth N− depth R, has been generalized in several directions; …

Let R be a Cohen–Macaulay local ring of dimension one with a canonical module K R. Let I
be a faithful ideal of R. We explore the problem of when I⊗ RI∨ is torsionfree, where I∨ …

A semidualizing module is a generalization of Grothendieck's dualizing module. For a local
Cohen-Macaulay ring $ R $, the ring itself and its canonical module are always realized as …

We obtain various characterizations of commutative Noetherian local rings (R, m) in terms of
homological dimensions of certain finitely generated modules. Our argument has a series of …

The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a
module, is one of the longstanding conjectures in ring theory. Although it is still open, there …

In this paper we consider a question of Roger Wiegand, which is about tensor products of
finitely generated modules that have finite projective dimension over commutative …