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- W2593040408 abstract "Let R be a ring with identity, X be the set of all nonzero, nonunits of R and G be the group of all units of R. A ring R is called unit-duo ring if <TEX>$[x]_{ell}=[x]_r$</TEX> for all <TEX>$x{in}X$</TEX> where <TEX>$[x]_{ell}={ux{mid}u{in}G}$</TEX> (resp. <TEX>$[x]_r={xu{mid}u{in}G}$</TEX>) which are equivalence classes on X. It is shown that for a semisimple unit-duo ring R (for example, a strongly regular ring), there exist a finite number of equivalence classes on X if and only if R is artinian. By considering the zero divisor graph (denoted <TEX>${tilde{Gamma}}(R)$</TEX>) determined by equivalence classes of zero divisors of a unit-duo ring R, it is shown that for a unit-duo ring R such that <TEX>${tilde{Gamma}}(R)$</TEX> is a finite graph, R is local if and only if diam(<TEX>${tilde{Gamma}}(R)$</TEX>) = 2." @default.
- W2593040408 created "2017-03-16" @default.
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- W2593040408 date "2016-11-30" @default.
- W2593040408 modified "2023-09-23" @default.
- W2593040408 title "UNIT-DUO RINGS AND RELATED GRAPHS OF ZERO DIVISORS" @default.
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- W2593040408 doi "https://doi.org/10.4134/bkms.b150684" @default.
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