SOME IMPROVED RESULTS ON B–RINGS

Amir M. Rahimi

Abstract. In this work, all rings are commutative rings with identity. Let R be a ring and J(R) denote the Jacobson radical of R. A sequence (a_{1}, a_{2}, …, a_{s}, a_{s+1}), s ≥ 1, of elements in R is said to be unimodular if 1 is in the ideal (a_{1}, a_{2}, …, a_{s}, a_{s+1}). A ring R is said to be a B–ring, whenever for any unimodular sequence (a_{1}, a_{2}, …, a_{s}, a_{s+1}), s ≥ 2, of elements in R with (a_{1}, a_{2}, …, a_{s−1}) ⊄ J(R), there exists an element b in R such that (a_{1}, a_{2}, …, a_{s} + ba_{s+1}) = R. Besides some other results, two basic facts about B–rings will be proved and they are: 1) R is a B–ring if and only if for any unimodular sequence (a_{1}, a_{2}, a_{3}) of elements in R with a_{1} not in J(R), there exists an element b in R such that (a_{1}, a_{2} + ba_{3}) = R, and 2) A homomorphic image of a B–ring is again a B–ring. At the end, some applications of these results will be discussed.

1. Introduction. The main purpose of this paper which will be discussed in section 2 of this work, is to provide a simpler alternative proof to some results of section 2 in [1] and also to generalize some results of that section in [1].

Throughout this paper all rings are commutative rings with identity. For a ring R, let J(R) denote the Jacobson radical of R and define a sequence (a_{1}, a_{2}, …, a_{s}, a_{s+1}), s ≥ 1, of elements in R to be unimodular, whenever 1 is in the ideal (a_{1}, a_{2}, …, a_{s}, a_{s+1}). Notice that without having any confusion in the context, parentheses will be used to show both the sequence (a_{1}, a_{2}, …, a_{s}, a_{s+1}) of elements in R and the ideal (a_{1}, a_{2}, …, a_{s}, a_{s+1}) generated by a_{1}, a_{2}, …, a_{s}, a_{s+1} ∈ R. A ring R is said to be a B–ring, whenever for any unimodular sequence (a_{1}, a_{2}, …, a_{s}, a_{s+1}), s ≥ 2, of elements in R with (a_{1}, a_{2}, …, a_{s−1}) ⊄ J(R), there exists an element b in R such that (a_{1}, a_{2}, …, a_{s} + ba_{s+1}) = R. For more information on B–rings see [1] and also for a generalization of B–rings (B–type–rings) see the dissertation of the author [2].

2. Main Results.

\underline{Theorem 1}. Let (a_{1}, a_{2}, …, a_{s}, a_{s+1}), s ≥ 1, be a unimodular sequence in R, then (a_{1}, a_{2}, …, a_{s} + a_{s+1}) = R, whenever a_{s} or a_{s+1} is in J(R).

\underline{Proof}. Here we just give a proof to the case when a_{s} ∈ J(R) and the proof of the other case which is similar, is left to the reader. Assume A = (a_{1}, a_{2}, …, a_{s} + a_{s+1}) ≠ R. Then there exists a maximal ideal M of R such that A ⊂ M. Now a_{s} ∈ J(R) ⊂ M and a_{s} + a_{s+1} ∈ M imply a_{s+1} ∈ M and this makes R = (a_{1}, a_{2}, …, a_{s+1}) ⊂ M, which is a contradiction.

\underline{Theorem 2}. A ring R is a B–ring if and only if for any unimodular sequence (a_{1}, a_{2}, a_{3}) in R with a_{1} ∉ J(R), there exists an element b ∈ R such that (a_{1}, a_{2} + ba_{3}) = R.

\underline{Proof}. The necessary part is quite clear. To prove the sufficient part, let (a_{1}, a_{2}, …, a_{s}, a_{s+1}), s ≥ 2, be a unimodular sequence in R with the condition (a_{1}, a_{2}, …, a_{s−1}) ⊄ J(R). Without loss of generality, assume a_{1} ∉ J(R). Now, 1 ∈ (a_{1}, a_{2}, …, a_{s}, a_{s+1}) implies 1 = ∑_{i=1}^{s+1} a_{i}x_{i} for some x_{1}, x_{2}, …, x_{s}, x_{s+1} ∈ R. Thus, 1 ∈ (a_{1}, a_{s}, l), where l = a_{2}x_{2} +a_{3}x_{3} +⋯+a_{s−1}x_{s−1} +a_{s+1}x_{s+1}. Now by the hypothesis there exists b ∈ R such that 1 ∈ (a_{1}, a_{s} + bl) ⊂ (a_{1}, a_{2}, …, a_{s−1}, a_{s} + bx_{s+1}a_{s+1}).

\underline{Theorem 3}. A homomorphic image of a B–ring is a B–ring.

\underline{Proof}. Let A be a nonzero proper ideal of R and ϕ: R → R/A be the canonical epimorphism of rings. Assume (a_{1}+A, a_{2}+A, …, a_{s}+A, a_{s+1}+A) is a unimodular sequence in R/A with (a_{1} + A, a_{2} + A, …, a_{s−1} + A) ⊄ J(R/A), where s ≥ 2. 1 + A ∈ (a_{1} + A, a_{2} + A, …, a_{s} + A, a_{s+1} + A) implies 1 = ∑_{i=1}^{s+1} a_{i}x_{i} + a for some a ∈ A and x_{1}, x_{2}, …, x_{s}, x_{s+1} ∈ R. Now 1 ∈ (a_{1}, a_{2}, …, a_{s}, a_{s+1}x_{s+1}+a) and since ϕ(J(R)) is contained in J(R/A), then by the hypothesis there exists b ∈ R such that (a_{1}, a_{2}, …, a_{s} + bx_{s+1}a_{s+1} + ba) = R, which implies (a_{1} + A, a_{2} + A, …, a_{s} + bx_{s+1}a_{s+1} + A) = R/A.

\underline{Remark}. By applying Theorem 2 in the proof of the above theorem, it suffices to make the argument only for the unimodular sequences of the form (a_{1} + A, a_{2} + A, a_{3} + A) with a_{1} + A ∉ J(R/A).

3. Some Applications of the Above Results. For a ring R and any element a ∈ R, let Z(a) denote the set of all maximal ideals of R, where each contains the element a. A ring R is said to be Bezoutian or an F –ring, if every finitely generated ideal in R is a principal ideal in R.

\underline{Remark}. R in Lemma 2.2 of [1] does not need to be an F –ring and actually, as a stronger alternative of this lemma, we can apply Theorem 2 above. Consequently, in Theorem 2.4 of [1] there is no need to show or mention that R is an F –ring. Also, we exclude the Bezoutian condition from the hypothesis of Theorem 2.3 in [1] and state it in the following general form.

\underline{Theorem 4}. If R is a ring which satisfies the condition that for every a, c ∈ R with a ∉ J(R), there exists an r ∈ R such that Z(R) = Z(a) − Z(c), then R is a B–ring.

\underline{Proof}. The proof follows from Theorem 2 above and the argument in the proof of Theorem 2.3 in [1].

Next, by applying Theorem 3 above, a simpler alternative proof can be provided to Theorem 2.6 and to the necessity part of Lemma 2.1 in [1].

\underline{Theorem 5}. Let D be an integral domain, K its quotient field. Let R = {(a_{1}, a_{2}, …, a_{k}, a, a, …) | a_{i} ∈ K, a ∈ D}, where k is a nonnegative integer (k may be different for distinct elements in R). The operations in R are component–wise addition and multiplication. If R is a B–ring, then D is a B–domain.

\underline{Proof}. Since D is a homomorphic image of R under the mapping given by (a_{1}, a_{2}, …, a_{k}, a, a, …) ↦ a, the proof is immediate by Theorem 3.

\underline{Remark}. By Theorem 2 above, the illustration of the proof of Theorem 2.6 in [1] is actually a complete proof of that theorem.

Finally, as an application of Theorem 3 to the necessity part of Lemma 2.1 in [1], which states “for any nonzero proper ideal A ⊂ J(R), R is a B–ring if and only if R/A is a B–ring,” we can conclude that R/A is a B–ring for any nonzero proper ideal A of R, whenever R is a B–ring.

\underline{References}

1. M. Moore and A. Steger, “Some Results on Completability in Commutative Rings,” Pacific Journal of Mathematics, 37 (1971), 453–460.

2. A. M. Rahimi, Some Results on Stable Range in Commutative Rings, Ph. D. dissertation, 1993, University of Texas at Arlington.

Amir M. Rahimi 901 Carro Dr. # 4 Sacramento, CA 95825

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