Ding modules and dimensions over formal triangular matrix rings

2019/12/15 by Lixin Mao, Mao, Lixin · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1912.06968

Abstract

Let T=\biggl(\beginmatrix A&0 U&B \endmatrix\biggr) be a formal triangular matrix ring, where A and B are rings and U is a (B, A)-bimodule. We prove that: (1) If U_A and _B U have finite flat dimensions, then a left T-module \biggl(\beginmatrix M₁ M₂\endmatrix\biggr)_φ^M is Ding projective if and only if M₁ and M₂/\rm im(φ^M) are Ding projective and the morphism φ^M is a monomorphism. (2) If T is a right coherent ring, _BU has finite flat dimension, U_A is finitely presented and has finite projective or FP-injective dimension, then a right T-module (W₁, W₂)_φ_W is Ding injective if and only if W₁ and ker(\widetildeφ_W) are Ding injective and the morphism \widetildeφ_W is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a T-module.

Cited by

• X-Gorenstein projective and Y-Gorenstein injective modules over tensor rings

Related

• Relative Gorenstein dimensions over triangular matrix rings
• Gorenstein homological dimensions of modules over triangular matrix rings
• Ding projective dimension of complexes
• n-cotorsion pairs over formal triangular matrix rings
• Gorenstein projective modules and recollements over triangular matrix rings