On the probability of zero divisor elements in group rings. (English) Zbl 1518.16026
Let \(R\) be a non-trivial finite commutative ring with identity and \(G\) be a finite group. Let \(P(RG)\) denote the probability that the product of two randomly chosen elements of the group ring \(RG\) is zero. The author proves that \(P(RG) \geq 1/4\) if and only if \(RG\) is isomorphic to one of the following rings: \( \mathbb{Z}_2 C_2\), \(\mathbb{Z}_3 C_2\), \(\mathbb{Z}_2 C_3\) (Theorem 3.10). Furthermore, there are established some upper and lower bounds for \(P(RG)\) (Lemmas 3.8, 3.9). In particular, the author obtains some formulas for calculations of \(P(RG)\) in the case, where \(R\) is a finite field and \(|G| \leq 4\) (Theorems 3.1, 3.2, 3.3, 3.4).
Reviewer: Anatolii Tushev (Dnipro)
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