Smarandache Vertices of the Graphs Associated to the Commutative Rings |
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Authors: | A. M. Rahimi |
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Affiliation: | 1. School of Mathematics , Institute for Research in Fundamental Sciences (IPM) , Tehran , Iran amrahimi@ipm.ir |
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Abstract: | Let R be a commutative ring with identity 1 ≠ 0. A nonzero element a in R is said to be a Smarandache zero-divisor if there exist three different nonzero elements x, y, and b (≠ a) in R such that ax = ab = by = 0, but xy ≠ 0. We will generalize this notion to the Smarandache vertex of an arbitrary simple graph and characterize the Smarandache zero-divisors of commutative rings (resp. with respect to an ideal) via their associated zero-divisor graphs. We illustrate them with examples and prove some interesting results about them. |
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Keywords: | Smarandache vertex of a graph Smarandache zero-divisor Zero-divisor graph of a ring (with respect to an ideal) |
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