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STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING

Year 2017, , 164 - 179, 17.01.2017
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. 

References

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  • 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.
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  • 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.
Year 2017, , 164 - 179, 17.01.2017
https://doi.org/10.24330/ieja.296324

Abstract

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.
There are 32 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Articles
Authors

A. Karimi Mansoub This is me

A Moussavi

M Habibi This is me

Publication Date January 17, 2017
Published in Issue Year 2017

Cite

APA Mansoub, A. K., Moussavi, A., & Habibi, M. (2017). STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING. International Electronic Journal of Algebra, 21(21), 164-179. https://doi.org/10.24330/ieja.296324
AMA Mansoub AK, Moussavi A, Habibi M. STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING. IEJA. January 2017;21(21):164-179. doi:10.24330/ieja.296324
Chicago Mansoub, A. Karimi, A Moussavi, and M Habibi. “STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING”. International Electronic Journal of Algebra 21, no. 21 (January 2017): 164-79. https://doi.org/10.24330/ieja.296324.
EndNote Mansoub AK, Moussavi A, Habibi M (January 1, 2017) STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING. International Electronic Journal of Algebra 21 21 164–179.
IEEE A. K. Mansoub, A. Moussavi, and M. Habibi, “STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING”, IEJA, vol. 21, no. 21, pp. 164–179, 2017, doi: 10.24330/ieja.296324.
ISNAD Mansoub, A. Karimi et al. “STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING”. International Electronic Journal of Algebra 21/21 (January 2017), 164-179. https://doi.org/10.24330/ieja.296324.
JAMA Mansoub AK, Moussavi A, Habibi M. STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING. IEJA. 2017;21:164–179.
MLA Mansoub, A. Karimi et al. “STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING”. International Electronic Journal of Algebra, vol. 21, no. 21, 2017, pp. 164-79, doi:10.24330/ieja.296324.
Vancouver Mansoub AK, Moussavi A, Habibi M. STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING. IEJA. 2017;21(21):164-79.