On perspective Abelian groups
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.
math.GR↗
arXiv subjects
Grigore Calugareanu
Publications and source records attributed to Grigore Calugareanu.
The formula ABA=Tr(AB)A for matrices
We prove that this formula characterizes the square matrices over commutative rings for which all 2 x 2 minors equal zero.
math.RA↗
On matrix invertible extensions over commutative rings
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.
math.AC↗
On idempotent stable range one matrices
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.
math.RA↗
Similarity for zero-square matrices
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.
math.RA↗
On unit stable range one matrices
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.
math.RA↗
On stable range one matrices
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.
math.RA↗