On generalized probability in finite commutative rings

Abstract

Let $R$ be a finite commutative ring with unity and $x\in R$. We study the probability that the product of two randomly chosen elements (with replacement) of $R$ equals $x$. We denote this probability by $Prob_x (R)$. We determine some bounds for this probability and also obtain some characterizations of finite commutative rings based on this probability. Moreover, we determine the explicit computing formulas for $Prob_x (R)$ when $R=\mathbb{Z}_m\times \mathbb{Z}_n$.

Keywords

• Probability
• finite commutative ring
• Euler-phi function
• simple graph
• zero-divisor graph
• complete graph
• local ring

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