Structures Related to Right Duo Factor Rings
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We study the structure of rings whose factor rings modulo nonzero proper ideals are right duo; such rings are called right FD. We first see that this new ring property is not left-right symmetric. We prove for a non-prime right FD ring R that R is a subdirect product of subdirectly irreducible right FD rings; and that R/N∗(R) is a subdirect product of right duo domains, and R/J(R) is a subdirect product of division rings, where N∗(R) (J(R)) is the prime (Jacobson) radical of R. We study the relation among right FD rings, division rings, commutative rings, right duo rings and simple rings, in relation to matrix rings, polynomial rings and direct products. We prove that if a ring R is right FD and 0 ̸= e2 = e ∈ R then eRe is also right FD, examining that the class of right FD rings is not closed under subrings.
Abstract
We study the structure of rings whose factor rings modulo nonzero proper ideals are right duo; such rings are called right FD. We first see that this new ring property is not left-right symmetric. We prove for a non-prime right FD ring R that R is a subdirect product of subdirectly irreducible right FD rings; and that R/N∗(R) is a subdirect product of right duo domains, and R/J(R) is a subdirect product of division rings, where N∗(R) (J(R)) is the prime (Jacobson) radical of R. We study the relation among right FD rings, division rings, commutative rings, right duo rings and simple rings, in relation to matrix rings, polynomial rings and direct products. We prove that if a ring R is right FD and 0 ̸= e2 = e ∈ R then eRe is also right FD, examining that the class of right FD rings is not closed under subrings.
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