n-perfectness in pullbacks

On Gorenstein global dimension in trivial ring extensions

“…Proof. Note that G.gldim(R) = sup{Gid(M )|M an R − module} (by [4,Theorem 1.1]). Thus, using [14, Theorem 2.22], G.gldim(R) ≤ n ⇔ Ext i (I, M ) = 0 for each i > n and for any injective module I and each module M .…”

“…Hence, T or i R⋉E (K, R) = 0 for all i > 0. Thus, using [13, Theorem 4.9], pd [4,Corollary 2.7]). Consequently, pd R⋉E (I) ≤ G.gldim(R) + n. Thus, from Lemma 2.7, we obtain the desired result.…”

“…Recall that the Gorenstein homological theory starts in the sixties with Auslander and Bridger [1,2] over commutative Noetherian rings and developed, several decades later, by Enochs, Jenda, Christensen, Holm, Yassemi and others (see [3,4,5,9,10,11,14]).…”

“…Recently in [4], the authors started the study of global Gorenstein dimensions of rings, which are called, for a commutative ring R, Gorenstein global projective, injective, and weak dimensions of R, denoted by GP D(R), GID(R), and G.wdim(R), respectively, and, respectively, defined as follows:…”

“…The homological behavior and structure of the R ⋉ E-module R has an importance counterpart in the determination of the Gorenstein and classical dimensions of the ring R ⋉ E. Recall that (see [4,Proposition 2.6])…”