Remark on Bi-Ideals and Quasi-Ideals of Variants of Regular Rings

Abstract: Problem statement: Every quasi-ideal of a ring is a bi-ideal. In general, a bi-ideal of a ring need not be a quasi-ideal. Every bi-ideal of regular rings is a quasi-ideal, so bi-ideals and quasi-ideals of regular rings coincide. It is known that variants of a regular ring need not be regular. The aim of this study is to study bi-ideals and quasi-ideals of variants of regular rings. Approach: The technique of the proof of main theorem use the properties of regular rings and bi-ideals. Results: It shows that every bi-ideal of variants of regular rings is a quasi-ideal. Conclusion: Although the variant of regular rings need not be regular but every bi-ideal of variants of regular rings is a quasi-ideal.


INTRODUCTION
The notion of quasi-ideals in rings was introduced by (Steinfeld, 1953) while the notion of bi-ideals in rings was introduced much later. It was actually introduced (Lajos and Sza'sz, 1971).
For nonempty subsets A, B of a ring R, AB denotes the set of all finite sums of the form i i i i a b ,a A,b B ∈ ∈ ∑ . A subring Q of a ring R is called a quasi-ideal of R if RQ∩QR⊆Q and a bi-ideal of R is a subring B of R such that BRB⊆B. Every quasi-ideal of R is a bi-ideal. In general, bi-ideals of rings need not be quasi-ideals. See the following example. Consider the ring 4 (SU ( ), , ) + ⋅ of all strictly upper triangular 4×4 matrices over the field of real numbers under the usual addition and multiplication of matrices.

MATERIALS AND METHODS
An element a of a ring R is called regular if there exists x in S such that a = axa. A ring R is called regular if every element in R is regular. The following known result shows a sufficient condition for a bi-ideal of a ring to be a quasi-ideal.
Theorem 1: If B is a bi-ideal of a ring R such that every element of B is regular in R, then B is a quasiideal of R. In particular, if R is a regular ring, then every bi-ideal of R is a quasi-ideal.
Let R be a ring and a∈R. A new product ο defined on R by x o y = xay for all x, y∈R. Then (R, +, o) is a ring. We usually write (R, +, a) rather that (R, +, o) to make the element a explicit. The ring (R, +, a) is called a variant of R with respect to a. It is well-known that the variant of regular rings need not be regular ring (see (Kemprasit, 2002) and (Chinram, 2009).
Our aim is to prove that every bi-ideal of variants of regular rings is a quasi-ideal. In fact, the technique of the proof of Theorem 1 is helpful for our work. However, our proof is more complicated.

RESULTS
The following theorem is our main result.
We then deduce from (6) and (7) Thus we obtain from (9) and (10) (1) and (8) it follows that x BaRaB ∈ which implies that x∈B. This proves that RaB BaR B, ∩ ⊆ so B is a quasiideal of the ring (R, +, a).
Hence the theorem is proved.

DISCUSSION
It is known that every bi-ideal of regular rings is a quasi-ideal. However, although the variant of regular rings need not be a regular ring but every bi-ideal of variants of regular rings is a quasi-ideal.

CONCLUSION
Every bi-ideal of variants of regular rings is a quasi-ideal, so bi-ideals and quasi-ideals of variants of regular rings coincide.