On radically-symmetric ideals
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A ring R is called symmetric, if abc=0 implies acb=0 for a,b,c ∈ R. An ideal I of a ring R is called symmetric (resp. radically-symmetric) if R/I (resp. R/(기호)) is a symmetric ring. We first show that symmetric ideals and ideals which have the insertion of factors property are radically-symmetric. We next show that if R is a semicommutative ring, then T_n(R) and R[x]/(x^n) are radically-symmetric,where (x^n) is the ideal of R[x] generated by x^n. Also we give some examples of radically-symmetric ideals which are not symmetric. Connections between symmetric ideals of R and related ideals of some ring extensions are also shown. In particular we show that if R is a symmetric (or semicommutative) (α, δ)-compatible ring, then R[x; α, δ] is a radically-symmetric ring. As a corollary we obtain a generalization of [13].
Abstract
A ring R is called symmetric, if abc=0 implies acb=0 for a,b,c ∈ R. An ideal I of a ring R is called symmetric (resp. radically-symmetric) if R/I (resp. R/(기호)) is a symmetric ring. We first show that symmetric ideals and ideals which have the insertion of factors property are radically-symmetric. We next show that if R is a semicommutative ring, then T_n(R) and R[x]/(x^n) are radically-symmetric,where (x^n) is the ideal of R[x] generated by x^n. Also we give some examples of radically-symmetric ideals which are not symmetric. Connections between symmetric ideals of R and related ideals of some ring extensions are also shown. In particular we show that if R is a symmetric (or semicommutative) (α, δ)-compatible ring, then R[x; α, δ] is a radically-symmetric ring. As a corollary we obtain a generalization of [13].
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