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Abstract
In the current article, we are going to investigate a relationship between some abstract ring structures (weakly PP-rings, clean rings, uniquely clean rings and n-clean rings) along with the skew generalized power series rings A[[N, ϑ]], where A is one of the ring structures described above, (N, ≤) represents a strictly ordered monoid while ϑ : N → End(A) represents a monoid homomorphism. We shall propose unified extensions of the above-mentioned ring structures by employing specific conditions along with their proofs.
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References
- Anderson, D.D. and Camillo F.P., “Commutative rings whose elements are a sum of a unit and idempotent”, Comm. Algebra, 30(7) (2002), 3327-3336.
- Liu, Z., “Special properties of rings of generalized power series”, Comm. Algebra, 32(8) (2004), 3215-3226.
- Liu, Z. and Ahsan, J., “PP-rings of generalized power series”, Acta Math. Sinica, English series, 16(4) (2000), 573-578.
- Marks, G., Mazurek, R. and Ziembowski. M., “A unified approach to various generalization of Armendariz rings”, Bull. Aust. Math. Soc., 81(2010), 361-397.
- Mazurek, R. and Ziembowski, M., “On von Nuemann regular rings of skew generalized power series”, Comm. Algebra, 36(2008), 1855-1868.
- Nicholson, W.K., “Lifting idempotents and exchange rings”, Tran. Am. Math. Soc., 229(1997), 269-278.
- Nicholson, W.K. and Zhou, Y., “Rings in which elements are uniquely the sum of an idempotent and a unit”, Glasg. Math. J., 46 (2004), 227-236.
- Paykan, K. and Moussavi, A., “Quasi-Armendariz generalized power series rings”, J. Algebra Appl. 15(5) (2016), 1650086, 38 p.
- Paykan, K. and Moussavi, A., “Baer and quasi Baer properties of skew generalized power series rings”, Comm. Algebra, 44(4) (2016), 1615-1635.
- Paykan, K. and Moussavi, A., “Some results on skew generalized power series rings”, Taiwanese Journal of Mathematics, 21 (2017), 11-26.
- Salem, R.M., “Noncommutative clean rings of generalized power series”, J. Egypt. Math. Soc., 14(2) (2006), 149-158.
- Singh, A.B. and Dixit, V.N., “Unification of extension of zip rings”, Acta Universitatis Sapientiae of Mathematics, 2(2) (2012), 168-181.
- Xiao, G. and Tong, W., “n-clean rings”, Algebra Coll., 13 (4) (2006), 599-606
References
Anderson, D.D. and Camillo F.P., “Commutative rings whose elements are a sum of a unit and idempotent”, Comm. Algebra, 30(7) (2002), 3327-3336.
Liu, Z., “Special properties of rings of generalized power series”, Comm. Algebra, 32(8) (2004), 3215-3226.
Liu, Z. and Ahsan, J., “PP-rings of generalized power series”, Acta Math. Sinica, English series, 16(4) (2000), 573-578.
Marks, G., Mazurek, R. and Ziembowski. M., “A unified approach to various generalization of Armendariz rings”, Bull. Aust. Math. Soc., 81(2010), 361-397.
Mazurek, R. and Ziembowski, M., “On von Nuemann regular rings of skew generalized power series”, Comm. Algebra, 36(2008), 1855-1868.
Nicholson, W.K., “Lifting idempotents and exchange rings”, Tran. Am. Math. Soc., 229(1997), 269-278.
Nicholson, W.K. and Zhou, Y., “Rings in which elements are uniquely the sum of an idempotent and a unit”, Glasg. Math. J., 46 (2004), 227-236.
Paykan, K. and Moussavi, A., “Quasi-Armendariz generalized power series rings”, J. Algebra Appl. 15(5) (2016), 1650086, 38 p.
Paykan, K. and Moussavi, A., “Baer and quasi Baer properties of skew generalized power series rings”, Comm. Algebra, 44(4) (2016), 1615-1635.
Paykan, K. and Moussavi, A., “Some results on skew generalized power series rings”, Taiwanese Journal of Mathematics, 21 (2017), 11-26.
Salem, R.M., “Noncommutative clean rings of generalized power series”, J. Egypt. Math. Soc., 14(2) (2006), 149-158.
Singh, A.B. and Dixit, V.N., “Unification of extension of zip rings”, Acta Universitatis Sapientiae of Mathematics, 2(2) (2012), 168-181.
Xiao, G. and Tong, W., “n-clean rings”, Algebra Coll., 13 (4) (2006), 599-606