STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING

Year 2017, Volume: 21 Issue: 21, 164 - 179, 17.01.2017

A. Karimi Mansoub A Moussavi , M Habibi

https://doi.org/10.24330/ieja.296324

Abstract

Let R be an associative ring with an endomorphism σ and F ∪ {0}
the free monoid generated by U = {u1, . . . , ut} with 0 added, and M a factor
of F setting certain monomial in U to 0, enough so that, for some n, Mn = 0.
Then we can form the skew monoid ring R[M; σ]. An element of a ring R is
strongly clean if it is the sum of an idempotent and a unit that commute. In
this paper, we prove that P
g∈M rgg ∈ R[M; σ] is a strongly clean element, if
re or 1 − re is strongly π-regular in R. As a corollary, we deduce that if R is a
strongly π-regular ring, then the skew monoid ring R[M; σ] is strongly clean.
These rings is a new family of non-semiprime strongly clean skew monoid rings. 

Keywords

Skew monoid ring, strongly clean ring

References

• [1] P. Ara, Strongly π-regular rings have stable range one, Proc. Amer. Math. Soc.,
• 124(11) (1996), 3293–3298.
• [2] G. Azumaya, Strongly π-regular rings, J. Fac. Sci. Hokkiado Univ. Ser. I, 13
• (1954), 34–39.
• [3] G. F. Birkenmeier, J.-Y. Kim and J. K. Park, A connection between weak
• regularity and the simplicity of prime factor rings, Proc. Amer. Math. Soc.,122(1) (1994), 53–58.
• [4] W. D. Burgess and P. Menal, On strongly π-regular rings and homomorphisms
• into them, Comm. Algebra, 16(8) (1988), 1701–1725.
• [5] J. Chen and Y. Zhou, Strongly clean power series rings, Proc. Edinb. Math.Soc., 50(1) (2007), 73–85.
• [6] J. Chen, X. Yang and Y. Zhou, On strongly clean matrix and triangular matrix
• rings, Comm. Algebra, 34(10) (2006), 3659–3674.
• [7] F. Dischinger, Sur les anneaux fortement π-reguliers, C. R. Acad. Sci. Paris,
• Ser. A-B, 283(8) (1976), 571–573.
• [8] M. Habibi and A. Moussavi, Annihilator properties of skew monoid rings,
• Comm. Algebra, 42(2) (2014), 842–852.
• [9] Y. Hirano, Some studies on strongly π-regular rings, Math. J. Okayama Univ.,
• 20(2) (1978), 141–149.
• [10] I. Kaplansky, Topological representations of algebras II, Trans. Amer. Math.
• Soc., 68 (1950), 62–75
• 11] T.-K. Lee and Y. Zhou, Armendariz and reduced rings, Comm. Algebra, 32(6)
• (2004), 2287–2299.
• [12] A. R. Nasr-Isfahani and A
• . Moussavi, On a quotient of polynomial rings,
• Comm. Algebra, 38(2) (2010), 567–575.
• [13] W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269–278.
• [14] W. K. Nicholson, Strongly clean rings and Fitting’s lemma, Comm. Algebra, 27(8) (1999), 3583–3592.
• [15] L. H. Rowen, Finitely presented modules over semiperfect rings, Proc. Amer.
• [16] L. H. Rowen, Examples of semiperfect rings, Israel J. Math., 65(3) (1989),
• 273–283.
• [17] Z. Wang and J. Chen, On two open problems about strongly clean rings, Bull.
• Austral. Math. Soc., 70(2) (2004), 279–282.
• [18] R. B. Warfield, Jr., Exchange rings and decompositions of modules, Math.