Ding modules and dimensions over formal triangular matrix rings

Mao, Lixin

doi:10.4171/rsmup/100

Mao, Lixin

Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 1-22

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Résumé

Let $T = (\begin{matrix} A & 0 \\ U & B \end{matrix})$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$ -bimodule. We prove: (1) If $U_{A}$ and $_{B} U$ have finite flat dimensions, then a left $T$ -module ${(\begin{matrix} M_{1} \\ M_{2} \end{matrix})}_{φ^{M}}$ is Ding projective if and only if $M_{1}$ and $M_{2} / im (φ^{M})$ are Ding projective and the morphism $φ^{M}$ is a monomorphism. (2) If $T$ is a right coherent ring, $_{B} U$ has finite flat dimension, $U_{A}$ is finitely presented and has finite projective or $FP$ -injective dimension, then a right $T$ -module ${(W_{1}, W_{2})}_{φ_{W}}$ is Ding injective if and only if $W_{1}$ and $ker (\tilde{φ_{W}})$ are Ding injective and the morphism $\tilde{φ_{W}}$ is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a $T$ -module.

MR Zbl

DOI : 10.4171/rsmup/100

Classification : 16

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     author = {Mao, Lixin},
     title = {Ding modules and dimensions over formal triangular matrix rings},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {1--22},
     volume = {148},
     year = {2022},
     doi = {10.4171/rsmup/100},
     mrnumber = {4542370},
     zbl = {07673819},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/rsmup/100/}
}

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Mao, Lixin. Ding modules and dimensions over formal triangular matrix rings. Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 1-22. doi: 10.4171/rsmup/100

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