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Classes of commutative clean rings. (English) Zbl 1210.13007

Throughout \(R\) is a commutative ring with identity. The ring \(R\)is said to be clean if each element of \(R\) is the sum of a unit and an idempotent and \(R\) is weakly clean if each element of \(R\) is the sum or difference of a unit and an idempotent. Recently clean rings have received wide attention. This paper introduces the notion of an \(I\)-clean ring. Let \(I\) be an ideal of \(R\). Then \(R\) is said to be \(I\)-clean (resp., weakly \(I\)-clean) if for each \(a\in R\)there is an idempotent \(e\) such that \(a-e\) (resp., at least one of \(a+e\) or \(a-e\)) isa unit and \(ae\in I\). The paper characterizes \(I\)-clean rings for the ideals \(0\), \(\sqrt{0}\), \(J(R)\) and \(R\) in terms of ring theoretic properties, multiplicative Noetherian filters of ideals (a filter of idealsis multiplicative if it is closed under products and is Noetherian if every ideal belonging to the filter contains a finitely generated ideal belonging to the filter), and the frame \(\mathfrak{C}_{R}\) of multiplicative Noetherian filters.
We have the following theorems.
Theorem 1. The following are equivalent: (1) \(R\) is \(0\)-clean, (2) \(R\) is weakly \(0\)-clean, (3) \(R\) is von Neumann regular, and (4) for finitely generated ideals \(I\) and \(J\), \(\sqrt{I}=\sqrt{J}\) implies \(I=J\).
Theorem 2. The following are equivalent: (1) \(R\) is \(\sqrt{0}\)-clean, (2) \(R\) is weakly \( \sqrt{0}\)-clean, (3) \(R\)is\(\;\)zero dimensional, (4) for every finitely generated ideal \(I\) there is an idempotent \(e\) such that \(I^{n}=Re\) for some natural number \(n\), and (5) \(\mathfrak{C}_{R}\) is a zero-dimensional frame.
Theorem 3. The following are equivalent: (1) \(R\) is \(J(R)\)-clean, (2) \(R\) is weakly \(J(R)\)-clean, (3) \(R/J(R)\) is von Neumann regular and idempotents lift mod \(J(R)\), (4) for every finitely generated ideal \(I\) there is an idempotent \(e\) such that \(I+J=R\) iff \(eR\subseteq J\) for any finitely generated ideal \(J\), and (5) \(\mathfrak{C}_{R}\) is a projective frame.
Theorem 4. The following are equivalent: (1) \(R\) is clean and (2) \(\mathfrak{C}_{R}\) is feebly projective.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
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References:

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