The clique ideal property (Q1711550)

Many papers appear on the topic of graphs from commutative rings. The first ever graph construction from a commutative ring is the zero-divisor graph by \textit{I. Beck} [J. Algebra 116, No. 1, 208--226 (1988; Zbl 0654.13001)]. Afterwards, it was modified by \textit{D. F. Anderson} and \textit{P. S. Livingston} [J. Algebra 217, No. 2, 434--447 (1999; Zbl 0941.05062)]. This is a paper on the graph defined by Beck. The modified zero-divisor graph is well studied on various perspections of rings and graphs. For a commutative ring \(R\), one can form a graph \(\Gamma(R)^*\) (as defined by Beck) whose vertices are the zero divisors of \(R\) (including 0) and whose edges are the pairs \(\{a, b\}\) where \(ab = 0\) with \(a\neq b\). For this graph, a clique is a nonempty subset \(X\) such that \(ab = 0\) for all \(a\neq b\) in \(X\). If \(R\) is a finite ring, there is always a maximum clique of \(\Gamma(R)^*\) -- a clique \(X\) such that \(|X| = |Y|\) for all cliques \(Y\). A finite ring \(R\) has the clique ideal property if each maximum clique of \(\Gamma(R)^*\) is an ideal of \(R\). This paper deals with rings with clique ideal property of commutative rings. If \(R = S\oplus T\) where both \(S\) and \(T\) are finite rings with the clique ideal property and neither \(S\) nor \(T\) is a field, then \(R\) has the clique ideal property. The converse does not hold. For each positive integer \(n > 1,\) the ring \(R = \mathbb{Z}_n[x]/(x^2)\) is a finite ring with the clique ideal property. In contrast, \(\mathbb{Z}_n\) has the clique ideal property if and only if \(n\) is either a prime or a perfect square.