Regular sequences in the subrings of C(X)

Authors: FARIBORZ AZARPANAH, Delavar ESMAEILVANDI

Abstract: We show that the intermediate subalgebras between $C^*(X)$ and C(X) do not contain regular sequences with length $\geq$ 2. This shows that depth(A(X)) $\leq$ 1 for each intermediate subalgebra A(X) between $C^*(X)$ and C(X). Whenever an intermediate subalgebra A(X) is proper, i.e. A(X) $\neq$ C(X), we observe that the depth of A(X) is exactly 1. Using this, it turns out that depth($C^*(X)$) = 0 if and only if $X$ is a pseuodocompact almost $P$ -space. The regular sequences in the subrings of the form $I + \mathbb{R}$ of C(X), where $I$ is a $z$-ideal of C(X), are also investigated and we have shown that the length of regular sequences in such rings is at most 1. In contrast to the depth of intermediate subalgebras, we see that the depth of a proper subring of the form $I + \mathbb{R}$ may be zero. Finally, regular sequences of extension rings of C(X) are also studied and some examples of subrings of C(X) are given with depths different from the depth of C(X).

Keywords: Regular sequence, depth, almost P-space, $z$-ideal

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