Search

We reobtain and often refine prior criteria due to Kaplansky, McGovern, Roitman, Shchedryk, Wiegand, and Zabavsky--Bilavska and obtain new criteria for a Hermite ring to be an \textsl{EDR}. We mention three criteria: (1) a Hermite ring $R$ is an \textsl{EDR} iff for all pairs $(a,c)\in R^2$, the product homomorphism $U(R/Rac)\times U\bigl(R/Rc(1-a)\bigr)\to U(R/Rc)$ between groups of units is surjective; (2) a reduced Hermite ring is an \textsl{EDR} iff it is a pre-Schreier ring and for each $a\in R$, every zero determinant unimodular $2\times 2$ matrix with entries in $R/Ra$ lifts to a zero determinant matrix with entries in $R$; (3) a B\'{e}zout domain $R$ is an \textsl{EDD} iff for all triples $(a,b,c)\in R^3$ there exists a unimodular pair $(e,f)\in R^2$ such that $(a,e)$ and $(be+af,1-a-bc)$ are unimodular pairs. We use these criteria to show that each B\'{e}zout ring $R$ that is an $(SU)_2$ ring (as introduced by Lorenzini) such that for each nonzero $a\in R$ there exists no nontrivial self-dual projective $R/Ra$-module of rank $1$ generated by $2$ elements (e.g., all its elements are squares), is an \textsl{EDR}., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:2303.08413

A unimodular $2\times 2$ matrix $A$ with entries in a commutative ring $R$ is called weakly determinant liftable if there exists a matrix $B$ congruent to $A$ modulo $R\det(A)$ and $\det(B)=0$; if we can choose $B$ to be unimodular, then $A$ is called determinant liftable. If $A$ is extendable to an invertible $3\times 3$ matrix $A^+$, then $A$ is weakly determinant liftable. If $A$ is simple extendable (i.e., we can choose $A^+$ such that its $(3,3)$ entry is $0$), then $A$ is determinant liftable. We present necessary and/or sufficient criteria for $A$ to be (weakly) determinant liftable and we use them to show that if $R$ is a $\Pi_2$ ring in the sense of Part I (resp.\ is a pre-Schreier domain), then $A$ is simply extendable (resp.\ extendable) iff it is determinant liftable (resp.\ weakly determinant liftable). As an application we show that each $J_{2,1}$ domain (as defined by Lorenzini) is an elementary divisor domain., Comment: 13 pages. Part II of the splitting of arXiv:2303.08413

A unimodular $2\times 2$ matrix with entries in a commutative $R$ is called extendable (resp.\ simply extendable) if it extends to an invertible $3\times 3$ matrix (resp.\ invertible $3\times 3$ matrix whose $(3,3)$ entry is $0$). We obtain necessary and sufficient conditions for a unimodular $2\times 2$ matrix to be extendable (resp.\ simply extendable) and use them to study the class $E_2$ (resp.\ $SE_2$) of rings $R$ with the property that all unimodular $2\times 2$ matrices with entries in $R$ are extendable (resp.\ simply extendable). We also study the larger class $\Pi_2$ of rings $R$ with the property that all unimodular $2\times 2$ matrices of determinant $0$ and with entries in $R$ are (simply) extendable (e.g., rings with trivial Picard groups or pre-Schreier domains). Among Dedekind domains, polynomial rings over $\mathbb Z$ and Hermite rings, only the EDRs belong to the class $E_2$ or $SE_2$. If $as(R)\le 2$, then $R$ is an $E_2$ ring iff it is an $SE_2$ ring., Comment: 17 pages in the original format (18 pages in the archive format). arXiv admin note: substantial text overlap with arXiv:2303.08413

As a special case of perspective R-modules, an Abelian goup is called perspective if isomorphic summands have a common complement. In this paper we describe many classes of such groups.

For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and unit-semiprime rings. Secondly, given a Lie ideal $L$ of a ring $R$, we offer a criterion for $R$ to be $L$-semiprime. For a prime ring $R$, we characterizes Lie ideals $L$ of $R$ such that $R$ is $L$-semiprime. Moreover, $X$-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent), semiprime rings, regular rings, and subdirect products are studied., Comment: Comments welcome

We prove that this formula characterizes the square matrices over commutative rings for which all 2 x 2 minors equal zero.

We introduce the class E2 (resp. SE2) of commutative rings R with the property that each unimodular 2 x 2 matrix with entries in R extends to an invertible 3 x 3 matrix (resp. invertible 3 x 3 matrix whose (3, 3) entry is 0). Among noetherian domains of dimension 1, polynomial rings over Z or Hermite rings, only EDRs belong to the class. Using this, stable ranges, and units and projective modules interpretations, we reobtain (often refine) criteria due to Kaplansky, McGovern, Roitman, Shchedryk, Wiegand and obtain new criteria for a Hermite ring to be an EDR. For instance, we characterize Hermite rings which are EDRs by (1) means of equations involving unimodular triples and (2) surjectivity of some maps involving units of factor rings by principal ideals. We use these criteria to show that each Bezout ring R that is either a J2,1 domain or an (SU)2 ring (as introduced by Lorenzini) such that for each nonzero a 2 R there exists no nontrivial self-dual projective R/Ra-module of rank 1 generated by 2 elements (e.g., all its elements are squares), is an EDR, thus solving or partially solving problems raised by Lorenzini. Many other classes of rings are introduced and studied. To ease the reading, this version concentrates only on extendability properties of unimodular 2 x 2 matrices and hence does not consider the case n > 2., Comment: 44 pages, longer version aimed to be split into 4 shorter parts

Let T be an n by n zero-square matrix over a commutative unital ring R. We show that T is similar to a multiple of E_1n if R is a GCD domain and n = 2, if R is a GCD domain with 2 not zero divisor and n = 3, but there are matrices which are not similar to any multiple of E_1n whenever n greater or equal 4, over any commutative unital ring.

We characterize the unit stable range one 2x2 and 3x3 matrices over commutative rings. In particular, we characterize the 2x2 matrices which satisfy the Goodearl-Menal condition. For 2x2 integral matrices we show that the stable range one and the unit stable range one properties are equivalent, and, that the only matrix which satisfies the Goodearl-Menal condition is the zero matrix.

We characterize the idempotent stable range one $2\times 2$ matrices over commutative rings and in particular, the integral matrices with this property. Several special cases and examples complete the subject.

For 2 by 2 matrices over commutative rings, we prove a characterization theorem for left stable range 1 elements, we show that the stable range 1 property is left-right symmetric (also) at element level, we show that all matrices with one zero row (or zero column) over Bezout rings have stable range 1. Using diagonal reduction, we characterize all the 2 by 2 integral matrices which have stable range 1 and discuss additional properties including Jacobson Lemma for stable range 1 elements. Finally, we give an example of exchange stable range 1 integral 2 by 2 matrix which is not clean., Comment: 10 pages

If $M$ is an $R$-module, we study the submodules $K\leq M$ with the property that $K$ is invariant with respect to all monomorphisms $K\rightarrow M$. Such submodules are called \textsl{strictly invariant}. For the case of $% \mathbb{Z}$-modules (i.e. Abelian groups) we prove that in many situations these submodules are invariant with respect to all homomorphisms $% K\rightarrow M$, submodules which were called \textsl{strongly invariant}., Comment: preprint version; the final version will be published by Mediterranean J. of Math

We classify by numerical invariants the finite subgroups $H$ of a primary abelian group $G$ for which every homomorphism or monomorphism of $H$ into $G$, or every endomorphism of $H$, extends to an endomorphism of $G$. We apply these results to show that for finitely generated subgroups of general abelian groups, the extendibility of monomorphisms implies the extendibility of all homomorphisms., Comment: v2: major revision, the title was changed, the paper was shortened, some results were removed