On $J$-rigid rings

Abstract

Let $R$ be a ring with an endomorphism $\sigma$. We introduce the notion of $\sigma$-$J$-rigid rings as a generalization of $\sigma$-rigid rings, and investigate its properties. It is proved that a ring $R$ is $\sigma$-$J$-rigid if and only if $R[[x;\sigma]]$ is $\bar\sigma$-$J$-rigid, while the $\sigma$-$J$-rigid property is not Morita invariant. Moreover, we prove that every ring isomorphism preserves $J$-rigid structure, and several known results are extended.

Keywords

• Rigid rings,Reduced rings,Jacobson radical,$\sigma$-$J$-rigid rings,over-rings

References

1. [1] S.A. Amitsur, Radicals of polynomial rings, Canad. J. Math. 8, 355–361, 1956.
2. [2] J. Chen, X. Yang, and Y. Zhou, On strongly clean matrix and triangular matrix rings, Comm. Algebra. 34, 3659–3674, 2006.
3. [3] W. Chen, Polynomial rings over weak Armendariz rings need not be weak Armendariz, Comm. Algebra. 42, 2528–2532, 2014.
4. [4] J. Dorroh, Concerning adjunctions to algebras, Bull. Amer. Math. Soc. 38 (2), 85–88, 1932.
5. [5] M. Habibi and A. Moussavi, Annihilator Properties of Skew Monoid Rings, Comm. Algebra, 42 (2), 842–852, 2014.
6. [6] M. Habibi, A. Moussavi and S. Mokhtari, On skew Armendariz of Laurent series type rings, Comm. Algebra, 40 (11), 3999–4018, 2012.
7. [7] C.Y. Hong, N.K. Kim and T.K. Kwak, Ore extensions of baer and pp-rings, J. Pure Appl. Algebra, 151 (3), 215–226, 2000.
8. [8] C.Y. Hong, N.K. Kim, and T.K. Kwak, On skew Armendariz rings, Comm. Algebra, 31 (1), 2511–2528, 2003.

9. [9] D. Jordan, Bijective extensions of injective ring endomorphisms, J. Lond. Math. Soc. 2 (3), 435–448, 1982.
10. [10] J. Krempa, Some examples of reduced rings, Algebra Colloq. 3, 289–300, 1996.
11. [11] J. Matczuk, A characterization of σ-rigid rings, Comm. Algebra, 32 (11), 4333–4336, 2004.
12. [12] K. Paykan and M. Habibi, Further results on Skew Monoid Rings of a certain free monoid, Cogent Math. Stat. 5 (1), 1–12, 2018.
13. [13] M. Sanaei, S. Sahebi, and H.H. Javadi, $\alpha$-skew $J$-Armendariz rings, J. Math. Ext. 12 (1), 63–72, 2018.