Flat modules and coherent endomorphism rings relative to some matrices

Following O'Meara's result [Journal of Algebra and Its Applications, 13 (2014) 8], it follows that the block matrix A = B 0 0 0 M n+r (R), B M n (R), r 1, over a von Neumann regular separative ring R, is a product of idempotent matrices.Furthermore, this decomposition into idempotents of A also holds when B is an invertible matrix and R is a GE ring (defined by Cohn [New mathematical monographs: 3, Cambridge University Press, 2006]).As a consequence, it follows that if there exists an example of a von Neumann regular ring R over which the matrix A = B 0 0 0 M n+r (R), where B M n (R), r 1, cannot be expressed as a product of idempotents, then R is not separative, thus providing an answer to an open question whether there exists a von Neumann regular ring which is not separative.The paper concludes with a counter example related to an open question whether every totally nonnegative matrix is a product of nonnegative idempotent matrices.

Let M M be a right R R -module with S = End ( M R ) S={\rm{End}}\left({M}_{R}) . Given two cardinal numbers α \alpha and β \beta and a row-finite matrix A ∈ RFM β × α ( S ) A\in {{\rm{RFM}}}_{\beta \times \alpha }\left(S) , S M {}_{S}M is called injective relative to A A if every left S S -homomorphism from S ( β ) A {S}^{\left(\beta )}A to M M extends to one from S ( α ) {S}^{\left(\alpha )} to M M . It is shown that S M {}_{S}M is injective relative to A A if and only if the right R R -module M β ∕ A M α {M}^{\beta }/A{M}^{\alpha } is cogenerated by M M . S S is called left coherent relative to A ∈ S β × α A\in {S}^{\beta \times \alpha } if Ker ( S ( β ) S → S ( β ) S A ) \left({}_{S}S^{\left(\beta )}\to {}_{S}S^{\left(\beta )}A) is finitely generated. It is shown that S S is left coherent relative to A A if and only if M n ∕ A M α {M}^{n}/A{M}^{\alpha } has an add ( M ) {\rm{add}}\left(M) -preenvelope. As applications, we obtain the necessary and sufficient conditions under which M n ∕ A M α {M}^{n}/A{M}^{\alpha } has an add ( M ) {\rm{add}}\left(M) -preenvelope, which is monic (resp., epic, having the unique mapping property). New characterizations of left n n -semihereditary rings and von Neumann regular rings are given.

In this paper, we first introduce and study the notions of [Formula: see text]-flat modules. Then study the rings over which all modules are [Formula: see text]-flat (which is called [Formula: see text]-VN regular rings). In particular, we show that a ring [Formula: see text] is a [Formula: see text]-VN regular ring if and only if T[Formula: see text] is a von Neumann regular ring, if and only if [Formula: see text] is a reduced ring with Min[Formula: see text] compact. Finally, we obtain the Chase theorem for [Formula: see text]-coheret rings: a ring [Formula: see text] is [Formula: see text]-coherent if and only if any direct product of [Formula: see text] is [Formula: see text]-flat if and only if any direct product of flat [Formula: see text]-modules is [Formula: see text]-flat. Some examples are provided to compare with the known conceptions.

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.

We characterize Ding modules and complexes over Ding-Chen rings. We show that, over a Ding-Chen ring $R$, the Ding projective (respectively, Ding injective, respectively, Ding flat) $R$-modules coincide with the Gorenstein projective (respectively, Gorenstein injective, respectively, Gorenstein flat) modules, which, in turn, are noth\-ing more than modules appearing as a cycle of an exact complex of projective (respectively, injective, respectively, flat) modules. We prove a similar characterization for chain complexes of $R$-modules: a complex~$X$ is Ding projective (respectively, Ding injective, respectively, Ding flat) if and only if each component $X_n$ is Ding projective (respectively, Ding injective, respectively, Ding flat). Along the way, we generalize some results of Stovicek and Bravo, Gillespie and Hovey to obtain other interesting corollaries. For example, we show that, over any Noetherian ring, any exact chain complex with Gorenstein injective components must have all cotorsion cycle modules, that is, $Ext ^1_R(F,Z_nI) = 0$ for any such complex $I$ and flat module $F$. On the other hand, over any coherent ring, the cycles of any exact complex $P$ with projective components must satisfy $Ext ^1_R(Z_nP,A) = 0$ for any absolutely pure module~$A$.

A left R-module M is called almost F-injective, if every R-homomorphism from a finitely presented left ideal to M extends to a homomorphism of R to M. A right R-module V is said to be almost flat, if for every finitely presented left ideal I, the canonical map V ⊗ I → V ⊗R is monic. A ring R is called left almost semihereditary, if every finitely presented left ideal of R is projective. A ring R is said to be left almost regular, if every finitely presented left ideal of R is a direct summand of RR. We observe some characterizations and properties of almost F-injective modules and almost flat modules. Using the concepts of almost F-injectivity and almost flatness of modules, we present some characterizations of left coherent rings, left almost semihereditary rings, and left almost regular rings.

Assignable polynomials to linear systems over von Neumann regular rings

Almost slender rings and compact rings

Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

We show that every flat semimodule having a projective cover coincides with that cover up to isomorphism. Also, we prove that all (finitely generated) semimodules over a semiring [Formula: see text] are flat iff [Formula: see text] is a von Neumann regular ring.

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.

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

In part I we introduce vertex rings , which bear the same relation to vertex algebras (or VOAs) as commutative, associative rings do to commutative, associative algebras over C \mathbf {C} . We show that vertex rings are characterized by Goddard axioms . These include a generalization of the translation-covariance axiom of VOA theory that involves a canonical Hasse-Schmidt derivation naturally associated to any vertex ring. We give several illustrative applications of these axioms, including the construction of vertex rings associated with the Virasoro algebra. We consider some categories of vertex rings, and the rôle played by the center of a vertex ring. In part II we extend the theory of Pierce bundles associated to a commutative ring to the setting of vertex rings. This amounts to the construction of certain reduced étale bundles of vertex rings functorially associated to a vertex ring. We introduce von Neumann regular vertex rings as a generalization of von Neumann regular commutative rings; we obtain a characterization of this class of vertex rings as those whose Pierce bundles are bundles of simple vertex rings.

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this article, we study the regularity of left SF-rings and we prove the following: 1) if R is a left SF-ring whose all complement left (right) ideals are W-ideals, then R is strongly regular; 2) if R is a left SF-ring whose all maximal essential right ideals are GW-ideals, then R is regular.