2018/12/26 by Fatma Kaynarca, Kaynarca, Fatma, H. Melis Tekin Akcin +1
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #math.RA #msc:16S36 #msc:16U80 #msc:16W20
paper · pdf · doi:10.48550/arxiv.1812.10291
arxiv created 2018/12/26 · arxiv updated 2018/12/27
Symmetric rings were introduced by Lambek to extend usual commutative ideal theory in noncommutative rings. In this paper, we study symmetric rings over which Ore extensions are symmetric. A ring R is called strongly σ-symmetric if the skew polynomial ring R[x;σ] is symmetric. We consider some properties and extensions of strongly σ-symmetric rings. Then we show the relationship between strongly σ-symmetric rings and other classes of rings. We next argue the polynomial extensions over strongly σ-symmetric rings. Moreover, we prove that if R is a σ-rigid ring, then R[x]/(xn) is a strongly σ-symmetric ring, where σis an endomorphism of R, (xn) is the ideal generated by xn and n is a positive integer; and that if the classical left quotient ring Q(R) of R exists, then R is σ-symmetric if and only if Q(R) is strongly σ-symmetric.