Semicommutativity of the rings relative to prime radical
COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE, cilt.56, sa.4, ss.401-415, 2015 (ESCI, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 56 Sayı: 4
- Basım Tarihi: 2015
- Doi Numarası: 10.14712/1213-7243.2015.140
- Dergi Adı: COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE
- Derginin Tarandığı İndeksler: Emerging Sources Citation Index (ESCI), Scopus
- Sayfa Sayıları: ss.401-415
- Anahtar Kelimeler: semicommutative ring, P-semicommutative ring, prime radical of a ring
- Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
- Ankara Üniversitesi Adresli: Evet
Özet
In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called P-semicommutative. We prove that a ring R is P-semicommutative if and only if R[x] is P-semicommutative if and only if R[x, x(-1)] is P-semicommutative. Also, if R[[x]] is P-semicommutative, then R is P-semicommutative. The converse holds provided that P(R) is nilpotent and R is power serieswise Armendariz. For each positive integer n, R is P-semicommutative if and only if T-n (R) is P-semicommutative. For a ring R of bounded index 2 and a central nilpotent element s, R is P-semicommutative if and only if K-s (R) is P-semicommutative. If T is the ring of a Morita context (A, B, M, N,Psi, phi with zero pairings, then T is P-semicommutative if and only if A and B are P-semicommutative. Many classes of such rings are constructed as well. We also show that the notions of clean rings and exchange rings coincide for P-semicommutative rings.