Let $T=\biggl(\begin{matrix} A&0\\U&B\end{matrix}\biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We first give some computing formulas of projective, injective, flat and $FP$-injective dimensions of special left $T$-modules. Then we establish some formulas of (weak) global dimensions of $T$. It is proven that (1) If $U_{A}$ is flat and $_{B}U$ is projective, $lD(A)\neq lD(B)$, then $lD(T)={\rm max}\{lD(A),lD(B)\}$; (2) If $U_{A}$ and $_{B}U$ are flat, $wD(A)\neq wD(B)$, then $wD(T)={\rm max}\{wD(A),wD(B)\}$.
formal triangular matrix ring projective dimension injective dimension flat dimension FP-injective dimension
National Natural Science Foundation of China
11771202
formal triangular matrix ring projective dimension injective dimension flat dimension FP-injective dimension
11771202
Birincil Dil | İngilizce |
---|---|
Konular | Matematik |
Bölüm | Matematik |
Yazarlar | |
Proje Numarası | 11771202 |
Yayımlanma Tarihi | 1 Aralık 2022 |
Yayımlandığı Sayı | Yıl 2022 Cilt: 51 Sayı: 6 |