A Characterization of Commutative Clean Rings
نویسنده
چکیده
A commutative ring A is said to be clean if every element of A can be written as a sum of a unit and an idempotent. This definition dates back to 1977 where it was introduced by W. K. Nicholson [7]. In 2002, V. P. Camillo and D. D. Anderson [1] investigated commutative clean rings and obtained several important results. In [4] Han and Nicholson show that if A is a semiperfect ring, then A[Z2] is a clean ring. In this paper we generalize this argument (for commutative rings) and show that A[Z2] is clean if and only A is clean. We also show that if the group ring A[G] is a commutative clean ring, then G must be a torsion group. Our investigations lead us to introduce the class of 2-clean rings.
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