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 [a_{1},⋯,a_{n}] is
called a primitive row vector provided 1 ∈ (a_{1},⋯,a_{n}); a primitive row vector
[a_{1},⋯,a_{n}] is called completable provided there exists an n × n unimodular matrix
over R with first row a_{1},⋯,a_{n}. A ring R is called a Bring if given a primitive row
vector [α_{1},⋯,a_{n}],n ≧ S, and
there exists b ∈ R such that 1 ∈ (a_{1},⋯,α_{n−2},a_{n−1} + ba_{n}). Similarly, R is defined to
be a Strongly Bring ( SBring), if d ∈ (a_{1},⋯,a_{n}),n ≧ 3, and (a_{1},⋯,a_{n−2})⊈J(R)
implies that there exists b ∈ R such that d ∈ (a_{1},⋯,a_{n−2},a_{n−1} + ba_{n}).
In this paper it is proved that every primitive vector over a Bring is completable.
It is shown that the following are Brings: πregular rings, quasisemilocal rings,
Noetherian rings in which every (proper) prime ideal is maximal, and adequate rings.
In addition it is proved that R[X] is a Bring if and only if R is a completely
primary ring. It is then shown that the following are SBrings: quasilocal
rings, amy ring which is both an Hermite ring and a Bring, and Dedekind
domains. Finally, it is shown that R[X] is an SBring if and only if R is a
field.
