Several results about the multiplicities of indecomposable injectives in the minimal injective resolution of a ring exist in the literature. Mostly these apply to universal enveloping algebras of finite dimensional solvable Lie algebras, and to Gorenstein noetherian PI local rings. We unify these results and extend them to the much wider class of rings with Auslander dualizing complexes.

We introduce the notion of a duality pair and demonstrate how the left half of such a pair is “often” covering and preenveloping. As an application, we generalize a result by Enochs et al. on Auslander and Bass classes, and we prove that the class of Gorenstein injective modules—introduced by Enochs and Jenda—is covering when the ground ring has a dualizing complex.

Let $C$ be a semidualizing module over commutative Noetherian local ring $R$. In this paper we introduce new class of modules, namely $C$-canonical modules which are generalization canonical modules. It is shown that if the exists then and converse holds under special conditions. Also, characterization Gorenstein rings given via

For a numerical semigroup ring K[H] we study the trace of its canonical ideal. The colength this ideal is called residue H. This invariant measures how far H from being symmetric, i.e. Gorenstein ring. We remark that contains conductor ideal, and bounds for residue. 3-generated semigroups give explicit formulas Thus, in setting can classify those whose at most one (the nearly ones), show eventu...

Let P be a commutative Noetherian ring, ???? an ideal of which is generated by regular sequence length four, f element P, and P¯ the hypersurface ring P?(f). Assume that ????:f grade four Gorenstein P. We give resolution N P¯?????P¯ free P¯-modules. The built from differential graded algebra P?(????:f) P-modules, together with one homotopy map. In particular, we explicit form for matrix factori...

We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms maximal number rooks that can be placed polyomino. also characterize Gorenstein polyominoes and those which are on punctured spectrum, compute Cohen–Macaulay type any polyomino rectangles covering it. Though main results algebraic nature, all proofs combinatorial.

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $\omega$. Attached to each such ring R, when $\omega$ is an ideal, there are integers--the type the reduction number $\omega$--that provide valuable metrics express deviation R from being Gorenstein ring. In arXiv:1701.05592 and arXiv:1711.09480 we enlarged this list with degree bi-canonical degree. work extend wh...

Let $\Delta =\left(\begin{smallmatrix} A & {_AN_B}\\ {_BM_A} B \\\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_{A}N=0=N\otimes_{B}M$.We first study how to construct (complete) duality pairs of $\Delta$-modules using $A$-modules and $B$-modules, generalizing the result Mao (Comm. Algebra, 2020, 12: 5296--5310) about over triangular matrix ring. Moreover, we Gorenstein projective modu...

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence and thus satisfy stable Harbourne conjecture. In prime characteristic, can take any radical ideal a ring in regular ring, provided its symbolic powers are given saturations with maximal ideal. Although this property is not suitable for reduction to characteristic p, simi...

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study projective degrees rational maps . use a general framework determine map via computation multiplicity saturated special fiber ring. As specific applications, we provide explicit formulas all determined perfect ideals height two or Gorenstein three.