WEAKLY STABLE CONDITIONS FOR EXCHANGE RINGS

Let R be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in R is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.

A ring R is a Garcia ring provided that the product of two regular elements is unit-regular. We prove that every regular element in a Garcia ring R is the sum/difference of an idempotent and a unit. Furthermore, we prove that every regular element in a weak Garcia ring is the sum of an idempotent and a one-sided unit. These extend several known theorems on (one-sided) unit-regular rings to wider classes of rings with sum summand property.

Stable modules and a theorem of Camillo and Yu

Suppose that [Formula: see text] and [Formula: see text] are elements of a, not necessarily unital, ring [Formula: see text] such that [Formula: see text] is regular. We prove that if [Formula: see text], then [Formula: see text] implies [Formula: see text] if and only if [Formula: see text]. This salvages Theorem 17 in [R. B. Bapat, S. K. Jain, K. M. P. Karantha and M. D. Raj, Outer inverses: Characterization and applications, Linear Algebra Appl. 528 (2017) 171–184] and extends Theorem 1 in [S. K. Jain and K. M. Prasad, Right-left symmetry of [Formula: see text] in regular rings, J. Pure Appl. Algebra 133 (1998) 141–142] to non-unital rings. For any positive integer [Formula: see text], if [Formula: see text] for [Formula: see text] and [Formula: see text] is regular, then we prove that [Formula: see text] implies [Formula: see text]. For any positive integer [Formula: see text], if [Formula: see text] for [Formula: see text] are such that [Formula: see text] and [Formula: see text], then we prove that the regularity of any [Formula: see text] elements of the set [Formula: see text] implies the regularity of the remaining element. Let [Formula: see text] and [Formula: see text] be regular elements of a ring [Formula: see text]. We prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the set of all inner inverses of the regular element [Formula: see text] in [Formula: see text]. This extends Theorem 2.2 in [T. K. Lee, Unit-regularity of elements in rings, J. Algebra Appl. 22 (2023) 2350135] where the result is proved assuming [Formula: see text] to be semiprime. We also give some applications of Jain–Prasad symmetry result Theorem 1 in [S. K. Jain and K. M. Prasad, Right-left symmetry of [Formula: see text] in regular rings, J. Pure Appl. Algebra 133 (1998) 141–142] by giving new characterizations of unit-regular and special clean elements. This leads to quick proofs of some known results.

We characterize exchange rings having stable range one. An exchange ring R has stable range one if and only if for any regular a ∈ R, there exist an e ∈ E(R) and a u ∈ U(R) such that a = e + u and aR ⋂ eR = 0 if and only if for any regular a ∈ R, there exist e ∈ r.ann(a +) and u ∈ U(R) such that a = e + u if and only if for any a, b ∈ R, R/aR ≅ R/bR ⇒ aR ≅ bR.

Invertible and regular elements in rings with involution

The article describes the experience of applying various approaches to the recognition of some of the most common settlements planning structures, which include radial, regular and ring elements. At the first stage, it is necessary to identify typical properties and elements of the corresponding figures, then find a way to automate the search for such entities in an arbitrary road network of a settlement. To solve the second problem, well-known algorithms are used that are associated with the analysis of the uniformity of the shape of neighborhoods of a settlement. Modifications of the algorithm for searching for radial structural elements are proposed, and the author’s method of detecting ring elements is tested. The selected approaches are implemented in the form of a script in the Python programming language, which sequentially checks the road network given to the input for the presence of lattice or radial-ring planning structure elements. The technique was tested in fifty cities of the world. Verification of the results was carried out by comparing the response of the algorithm with the expert opinion of the research authors or literature on the topic. The accuracy of the classification was 80 %. The algorithm successfully coped with the reference examples of layouts, but experienced difficulties with their modifications, associated, for example, with the lack of closure of the ring elements of the radial-ring structure. The results can be demanded in problems of cartographic generalization, which require recognition and preservation of typical features of spatial objects.

On sets of elements in Rickart rings induced by partial orders

We investigate partial cancellation of modules and show that if an ideal I of an exchange ring R has stable range one, then A⊕B≌A⊕C implies B≌C for all A∈FP (I). The converse is true when R is a regular ring. For an ideal I of a regular ring, we also show that I has stable range one if and only if perspectivity is transitive in L(A) for all A∈ FP (I). These give nontrivial generalizations for unit-regularity.