Let $(R,\mathfrak {m}) $ be a commutative Noetherian local ring, and suppose $ R $ is
Cohen-Macaulay with canonical module $\omega_R $. We develop new tools for analyzing …

In this paper, we introduce generalized Gorenstein local (GGL) rings. The notion of GGL
rings is a natural generalization of the notion of almost Gorenstein rings, which can thus be …

Auslander-Reiten Conjecture for commutative Noetherian rings predicts that a finitely
generated module is projective when certain Ext-modules vanish. But what if those Ext …

Upper bound on the colength of the trace of the canonical module in dimension one | Archiv der
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We investigate a problem of when commutative local domains have a finite number of trace
ideals. The problem is left for the case of dimension one. In this paper, with a necessary …

Let $ R $ be a domain that is a complete local $\mathbb {k} $ algebra in dimension one. In
an effort to address the Berger's conjecture, a crucial invariant reduced type $ s (R) $ was …

We investigate the nearly Gorenstein property of a local ring defined by the maximal minors
of a specific $2\times n $ matrix with entries in the formal power series ring $ k [[X_1 …

In this paper, we study Noetherian local rings R having a finite number of trace ideals. We
proved that such rings are of dimension at most two. Furthermore, if the integral closure of …

For any finitely generated module M with non-zero rank over a commutative one
dimensional Noetherian local domain, the numerical invariant h (M) was introduced and …

Our aim throughout this thesis is to illuminate combinatorial and homological properties of
algebraic structures arising in combinatorial commutative algebra, combinatorics, and …