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The zero-divisor graph with respect to ideals of a commutative semiring. (English) Zbl 1162.16031

The zero-divisor graph of a commutative semiring is the simple graph whose vertex set is the set of non-zero zero divisors and an edge is drawn between two distinct vertices if their product is zero.
Let \(R\) be a commutative semiring with \(0\neq 1\) and \(\Gamma(R)\) its zero-divisor graph. For an ideal \(I\), the graph \(\Gamma_I(R)\) has its vertex set \(\{x\in R-I:xy\in I\) for an \(y\in R-I\}\) and two distinct vertices are adjacent if \(xy\in I\). It is obvious that: (i) If \(I=(0)\), then \(\Gamma_I(R)=\Gamma(R)\). (ii) If \(I\neq (0)\), then \(I\) is a prime ideal if and only if \(\Gamma_I(R)=\emptyset\). (iii) \(\Gamma_I(R)\) is connected with the diameter \(\leq 3\).
If \(I\) is a \(Q\)-ideal (i.e., there is a subset \(Q\) of \(R\) such that \(R=\bigcup\{q+I:q\in Q\}\) and \((q+I)\cap(q_2+I)\) is nonempty if and only if \(q=q_2)\), then \(\Gamma_I(R)\) and \(\Gamma(R/I)\) are empty simultaneously.
The girth of a graph \(\Gamma\) is the length of the shortest cycle in \(\Gamma\), if there are cycles, or is \(\infty\) otherwise. The author gives some properties of the girth of \(\Gamma(R)\) and of the girth of \(\Gamma_I(R)\).
The case of Noetherian semirings and the graphs of primary ideals in semirings are also considered.

MSC:

16Y60 Semirings
05C75 Structural characterization of families of graphs
16D25 Ideals in associative algebras
13A15 Ideals and multiplicative ideal theory in commutative rings
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