ON STRONGLY QUASI PRIMARY IDEALS - (:unkn) unknown (KOREAN MATHEMATICAL SOC, 2019)
In this paper, we introduce strongly quasi primary ideals which is an intermediate class of primary ideals and quasi primary ideals. Let R be a commutative ring with nonzero identity and Q a proper ideal of R. Then Q is called strongly quasi primary if ab is an element of Q for a, b is an element of R implies either a(2) is an element of Q or b( )(n)is an element of Q (a(n) is an element of Q or b(2) is an element of Q) for some n is an element of N. We give many properties of strongly quasi primary ideals and investigate the relations between strongly quasi primary ideals and other classical ideals such as primary, 2-prime and quasi primary ideals. Among other results, we give a characterization of divided rings in terms of strongly quasi primary ideals. Also, we construct a subgraph of ideal based zero divisor graph Gamma(I) (R) and denote it by Gamma(I)*(R), where I is an ideal of R. We investigate the relations between Gamma(I)*(R) and Gamma(I) (R). Further, we use strongly quasi primary ideals and Gamma(I)*(R) to characterize von Neumann regular rings.
Keywords: valuation domain; divided ring; strongly quasi primary ideal; zero divisor graph; ideal based zero divisor graph; ZERO-DIVISOR GRAPH