1. Computation of depth of factor rings of C(X).
- Author
-
Hesari, A. A. and Salehi, A. R.
- Subjects
- *
LOGICAL prediction , *LITERATURE - Abstract
It is known that the depth of every factor ring of C (X) module an ideal is at most 1. In this paper, we examine conditions under which the depth of factor rings of C (X) module closed ideals are either 0 or 1. Particularly, we show that the depth of factor ring C (X) / M A , A ⊆ X , is 0 (or equivalently this ring is classical i.e. its every element is unit or zerodivisor) if and only if A is an almost P -space completely separated from every zero-set disjoint from it. Using this, it has been confirmed that C (X) modulo the smallest z ∘ -ideal containing f ∈ C (X) is classical if and only if cl X int X Z (f) is an almost P -space completely separated from every zero-set disjoint from it. Also, it has been verified that X is a P -space if and only if for every ideal I ⊆ C (X) , the factor ring C (X) / I has depth zero. Finally, we present a counterexample to a conjecture about the depth of subrings of C (X) in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF