TY  - JOUR
T1  - On the equation  $a^{2}x=a$  in unital rings
AU  - Călugăreanu, Grigore
AU  - Pop, Horia F.
PY  - 2025
DA  - July
Y2  - 2024
DO  - 10.24330/ieja.1578725
JF  - International Electronic Journal of Algebra
JO  - IEJA
PB  - Abdullah HARMANCI
WT  - DergiPark
SN  - 1306-6048
SP  - 37
EP  - 51
VL  - 38
IS  - 38
LA  - en
AB  - In an arbitrary ring, the equation in the title defines the left strongly regular elements. Elements that are both left and right strongly regular are simply called strongly regular. If all elements of a ring are left strongly regular, then they are, in fact, strongly regular and this is the definition of strongly regular rings.We provide a characterization of when a left strongly regular element is indeed strongly regular, based on an intrinsic condition. While we show that it is not possible to give $2\times 2$ non-examples over $\mathbb{Z}$ or in certain matrix rings over $\mathbb{Z}_{n}$ for $n\in \{8,9,16\}$, we present two examples by George Bergman: a left strongly regular element that is not regular and a regular, left strongly regular element that is not strongly regular.Further, we prove results for left strongly regular (square) matrices over various types of rings and propose a conjecture, strongly supported by computational evidence: over commutative rings, left strongly regular matrices are strongly regular.
KW  - Left strongly regular element
KW  - strongly regular ring
KW  - 2x2 matrix
KW  - regular element
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UR  - https://doi.org/10.24330/ieja.1578725
L1  - https://dergipark.org.tr/en/download/article-file/4337315
ER  -