Vol. 37, No. 2, 1971

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Some results on completability in commutative rings

Marion Edward Moore and Arthur Steger

Vol. 37 (1971), No. 2, 453–460

In this paper, R always denotes a commutative ring with identity. The ideal of nHpotents and the Jacobson radical of the ring R are denoted by N(R) and J(R), respectively. The vector [a1,,an] is called a primitive row vector provided 1 (a1,,an); a primitive row vector [a1,,an] is called completable provided there exists an n × n unimodular matrix over R with first row a1,,an. A ring R is called a B-ring if given a primitive row vector [α1,,an],n S, and

(a1,⋅⋅⋅ ,an−2) ⁄⊆ J (R ),

there exists b R such that 1 (a1,n2,an1 + ban). Similarly, R is defined to be a Strongly B-ring ( SB-ring), if d (a1,,an),n 3, and (a1,,an2)⊈J(R) implies that there exists b R such that d (a1,,an2,an1 + ban).

In this paper it is proved that every primitive vector over a B-ring is completable. It is shown that the following are B-rings: π-regular rings, quasi-semi-local rings, Noetherian rings in which every (proper) prime ideal is maximal, and adequate rings. In addition it is proved that R[X] is a B-ring if and only if R is a completely primary ring. It is then shown that the following are SB-rings: quasi-local rings, amy ring which is both an Hermite ring and a B-ring, and Dedekind domains. Finally, it is shown that R[X] is an SB-ring if and only if R is a field.

Mathematical Subject Classification 2000
Primary: 13F05
Received: 30 October 1969
Published: 1 May 1971
Marion Edward Moore
Arthur Steger