Journal of Commutative Algebra

Semiclean rings and rings of continuous functions

Nitin Arora and S. Kundu

Full-text: Open access

Abstract

As defined by Ye [{\bf12}], a ring is semiclean if every element is the sum of a unit and a periodic element. Ahn and Anderson [{\bf1}] called a ring {weakly clean} if every element can be written as $u+e$ or $u-e$, where $u$ is a unit and $e$ an idempotent. A weakly clean ring is {semiclean}. We show the existence of semiclean rings that are not weakly clean. Every semiclean ring is $2$-clean. New classes of semiclean subrings of $\r$ and $\c$ are introduced and conditions are given when these rings are clean. Cleanliness and related properties of $C(X,A)$ are studied when $A$ is a dense semiclean subring of $\r$ or $\c$.

Article information

Source
J. Commut. Algebra, Volume 6, Number 1 (2014), 1-16.

Dates
First available in Project Euclid: 2 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.jca/1401715575

Digital Object Identifier
doi:10.1216/JCA-2014-6-1-1

Mathematical Reviews number (MathSciNet)
MR3215558

Zentralblatt MATH identifier
1294.16025

Subjects
Primary: 13A99: None of the above, but in this section 13B30: Rings of fractions and localization [See also 16S85] 16S60: Rings of functions, subdirect products, sheaves of rings 54C40: Algebraic properties of function spaces [See also 46J10]

Keywords
Clean rings Semiclean Weakly clean Rings of continuous functions

Citation

Arora, Nitin; Kundu, S. Semiclean rings and rings of continuous functions. J. Commut. Algebra 6 (2014), no. 1, 1--16. doi:10.1216/JCA-2014-6-1-1. https://projecteuclid.org/euclid.jca/1401715575


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References

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