Let T be a weak torsion class of left R-modules and n a positive integer. A left R-module M is called (T, n)-injective if Ext (C, M) = 0 for each (T, n + 1)-presented left R-module C; a right R-module M is called (T, n)-flat if Tor (M, C) = 0 for each (T, n +1)-presented left R-module C; a left R-module M is called (T, n)-projective if Ext (M, N) = 0 for each (T, n)-injective left R-module N; the ring R is called strongly (T, n)-coherent if whenever 0 → K → P → C → 0 is exact, where C is (T, n + 1)-presented and P is finitely generated projective, then K is (T, n)-projective; the ring R is called (T, n)-semihereditary if whenever 0 → K → P → C → 0 is exact, where C is (T, n + 1)-presented and P is finitely generated projective, then pd(K) ⩽ n → 1. Using the concepts of (T, n)-injectivity and (T, n)-flatness of modules, we present some characterizations of strongly (T, n)-coherent rings, (T, n)-semihereditary rings and (T, n)-regular rings.

中文翻译：

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强 $(\mathcal{T},n)$-相干环、$(\mathcal{T},n)$-半遗传环和 $(\mathcal{T},n)$-正则环

设 T 是左 R 模和 na 正整数的弱扭转类。如果对于每个 (T, n + 1)-呈现的左 R-模 C，如果 Ext (C, M) = 0，则左 R-模 M 被称为 (T, n)-内射；如果 Tor (M, C) = 0 对于每个 (T, n +1)-呈现的左 R-模数 C，则右 R-模数 M 被称为 (T, n)-平坦；如果 Ext (M, N) = 0 对于每个 (T, n)-射入左 R-模 N，则左 R-模 M 被称为 (T, n)-射影；如果每当 0 → K → P → C → 0 是精确的，其中 C 是 (T, n + 1)-呈现并且 P 是有限生成的射影，则环 R 被称为强 (T, n)-相干的，则 K 是 ( T, n)-投影；如果每当 0 → K → P → C → 0 是精确的，其中 C 是 (T, n + 1)-呈现并且 P 是有限生成的射影，则环 R 被称为 (T, n)-半遗传的，则 pd(K) ⩽ n → 1. 使用模的(T, n)-注入性和(T, n)-平坦度的概念，