On \(n\)-absorbing ideals and the \(n\)-Krull dimension of a commutative ring
Let \(R\) be a commutative ring and \(n\) a positive integer. In the paper under review, the authors introduce the \(n\)-Krull dimension of \(R\), denoted by \(\dim_n R\), which is the supremum of the lengths of chains of \(n\)-absorbing ideals of \(R\). Recall that for a positive integer \(n\), a proper ideal \(I\) of \(R\) is called an \(n\)-absorbing ideal if whenever \(x_1\cdots x_{n+1}\in I\) for \(x_1,\ldots,x_{n+1} R\), then there are \(n\) of the \(x_i\)'s whose product is in \(I\). It is clear that 1-absorbing ideals are just prime ideals. The authors extend several results of Krull dimension to \(n\)-Krull dimension and study the \(n\)-Krull dimension in several classes of commutative rings. For example, the \(n\)-Krull dimension of an Artinian ring is finite for every positive integer \(n\). In particular, if \(R\) is an Artinian ring with \(k\) maximal ideals and \(l(R)\) is the length of a composition series for \(R\), then dim\(_n R = l(R)-k\) for some positive integer \(n\). It is proved that a Noetherian domain \(R\) is a Dedekind domain if and only if dim\(_n R=n\) for every positive integer \(n\) if and only if dim\(_2 R=2\). It is also shown that Krull's (Generalized) Principal Ideal Theorem does not hold in general when prime ideals are replaced by \(n\)-absorbing ideals for some \(n > 1\).
scientific article; zbMATH DE number 6658981