Gorenstein homological dimensions of modules over triangular matrix rings

Abstract

Let A and B be rings, U a (B, A)-bimodule and T = ( A 0 U B ) be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over T , and discuss when a left T -module is strongly Gorenstein projective or strongly Gorenstein injective module.

Relative Gorenstein dimensions over triangular matrix rings

“…G C -projectives. Thus, G C −pd(M ) ≤ m.The following consequence of Theorem 5.3 extends [10, Proposition 2.8(1)] and[26, Theorem 2.7(1)] to the relative setting.Corollary 5.4 Let C = p(C 1 , C 2 ) be w-tilting, U C-compatible and M = M 1 M 2 ϕ M a T -module. If SG C2 − P D(B) < ∞, then G C −pd(M ) < ∞ if and only if G C1 −pd(M 1 ) < ∞ and G C2 −pd(M 2 ) < ∞.The following theorem gives an estimate of the left G C -projective global dimension of T .Theorem 5.5 Let C = p(C 1 , C 2 ) be w-tilting and U C-compatible.…”

“…Enochs, Izurdiaga, and Torrecillas [10] characterized when a left module over a triangular matrix ring is Gorenstein projective or Gorenstein injective under the "Gorenstein regular" condition. Under the same condition, Zhu, Liu, and Wang [26] investigated Gorenstein homological dimensions of modules over triangular matrix rings. Mao [25] studied Gorenstein flat modules over T (without the "Gorenstein regular" condition) and gave an estimate of the weak global Gorenstein dimension of T .…”

“…So, formal triangular matrix rings and modules over them have proved to be a rich source of examples and counterexamples. Some important Gorenstein notions over formal triangular matrix rings have been studied by many authors (see [30,10,26]). For example, Zhang [30] introduced compatible bimodules and explicitly described the Gorenstein projective modules over triangular matrix Artin algebra.…”

Relative Gorenstein Dimensions over Triangular Matrix Rings

“…Enochs, Izurdiaga, and Torrecillas [2] characterized when a left module over a triangular matrix ring is Gorenstein projective or Gorenstein injective under the "Gorenstein regular" condition. Under the same condition, Zhu, Liu, and Wang [3] investigated Gorenstein homological dimensions of modules over triangular matrix rings. Mao [4] studied Gorenstein flat modules over T (without the "Gorenstein regular" condition) and gave an estimate of the weak global Gorenstein dimension of T.…”

“…Therefore, formal triangular matrix rings and modules over them have proven to be a rich source of examples and counterexamples. Some important Gorenstein notions over formal triangular matrix rings have been studied by many authors (see [1][2][3]). For example, Zhang [1] introduced compatible bimodules and explicitly described the Gorenstein projective modules over triangular matrix Artin algebra.…”