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Generalized $$n$$-coherence. (English) Zbl 1041.16001
Let $$R$$ be any associative ring with identity element. A right module $$M$$ is said to be $$n$$-presented if there is an exact sequence $$F_n\to F_{n-1}\to\cdots\to F_1\to F_0\to M\to 0$$ with $$F_i$$, $$i=0,\dots,n$$, finite rank free. A left $$R$$-module $$Q$$ is called $$n$$-flat provided $$\text{Tor}_n^R(N,Q)=0$$ for all $$n$$-presented right $$R$$-modules $$N$$. A right $$R$$-module $$M$$ is said to be $$(\aleph,Q)$$-finitely generated if every subset $$T\subseteq M\otimes_RQ$$ with $$|T|<\aleph$$ is contained in $$N\otimes_RQ$$ for some finitely generated submodule $$N\leq M$$. Further, $$M$$ is called $$n$$-$$(\aleph,Q)$$-presented if there exists an exact sequence $$0\to K_n\to F_{n-1}\to\cdots\to F_1\to F_0\to M\to 0$$ with $$F_0, \dots,F_{n-1}$$ finite rank free and $$K_n$$ $$(\aleph ,Q)$$-finitely generated. Finally, a ring $$R$$ is said to be $$n$$-$$(\aleph,Q)$$-coherent if every $$n$$-presented right $$R$$-module is $$(n+1)$$-$$(\aleph,Q)$$-presented.
The author proves that the following conditions for a non-negative integer $$n$$ are equivalent: (i) the $$\aleph$$-product $$\prod_{i\in I}^\aleph Q=\{m\in Q^I\mid\text{supp}(m)<\aleph\}$$ is $$n$$-flat for every index set $$I$$; (ii) $$\prod_{i\in I}^\aleph Q_i$$ is $$n$$-flat for every index set $$I$$ and any family of $$n$$-flat modules $$Q_i\in\text{Gen}(Q)$$; (iii) $$R$$ is right $$n$$-$$(\aleph,Q)$$-coherent; (iv) $$\text{Tor}_n^R(N,\prod_{i\in I}^\aleph Q_i)\cong\prod_{i\in I}^\aleph\text{Tor}_n^R(N,Q_i)$$ for every $$n$$-presented right $$R$$-module $$N$$ and any family of left $$R$$-modules $$Q_i\in\text{Gen}(Q)$$ (Theorem 1.12).
Let $$R$$ be a ring and let $$\circ$$ denote the adjoint operation given by $$a\circ b=a+b-ab$$ for all $$a,b\in R$$. A ring $$R$$ is said to be a closure ring if it has an additional unary operation $$C$$ such that for all $$a,b\in R$$ it holds: (1) $$C(0)=0$$; (2) $$a\circ C(a)=C(a)$$; (3) $$C(C(a))=C(a)$$; (4) $$C(a)\circ C(b)=C(b)\circ C(a)$$; (5) $$C(a)\circ C(b)=C(a\circ b)\circ C(b)$$. If, moreover, $$a\circ b\circ C(a)=b\circ C(a)$$, then $$R$$ is said to be a strong closure ring.
The first part presents some basic properties of such rings and the second one deals with the relations between normal filters and closed ideals. The third paragraph relates the closure operation on the ring $$R$$ and on the matrix ring $$M_n(R)$$ and on the polynomial ring $$R[S]$$ ($$S$$ is a set of indeterminates), respectively. In the last item the closure operations defined in terms of the lattice of central idempotents are investigated.
##### MSC:
 16D40 Free, projective, and flat modules and ideals in associative algebras 16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
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