Uniformly $\pi$-regular rings and semigroups: a survey

Stojan Bogdanović, Miroslav Cirić, Tatjana Petković

The main aim of this paper is to give a survey of the most important structural properties of uniformly $\pi$-regular rings and semigroups. It is well-known that there are many similarities between certain types of semigroups and related rings. For example, we will see in Theorem 2.1 that the regularity of a semigroup can be characterized by means of the properties of its left and right ideals, and in the same way, the regularity of a ring can be characterized through its ring left and right ideals. On the other hand, there are many significant differences between the properties of certain types of semi groups and the properties of related rings. For example, many concepts such as the left, right and complete regularity and other, are different in Theory of semigroups, but they coincide in Theory of rings. One of the main goals of this paper is exactly to underline both the similarities and differences between related types of rings and semigroups. For that purpose many interesting results of Theory of rings or Theory of semi groups will be omitted here if they are not similar or essentially different than the corresponding result of another theory.