two-sided ideal

two-sided ideal

[′tü ¦sīd·əd ī′dēl]
(mathematics)
A two-sided ideal I is a sub-ring of a ring R where the products xy and yx are always in I for every x in R and y in I.
References in periodicals archive ?
(3) By two-sided ideal or, simply, ideal, we mean a nonempty subset of an ordered AG-groupoid S which is both left and right ideal of S.
A right ideal (left ideal, two-sided ideal) is a non-empty language L over an alphabet [summation] such that L = L[summation]* (L = [summation]* L, L = [summation]* L[summation]*).
Similarly we can show that [N (L) [union] N (S) N (L)] is a neutrosophic two-sided ideal of N(S).
While the ring [mathematical expression not reproducible] is a commutative ring, so every ideal in [mathematical expression not reproducible] is two-sided ideal, the skew polynomial ring [mathematical expression not reproducible] is noncommutative.
Let R be a non-commutative prime ring, a, b [member of] R, I [a.sub.n] two-sided ideal of R, n [greater than or equal to] 1 a fixed integer such that [a[[r.sub.1], [r.sub.2]] + [[r.sub.1], [r.sub.2]]b, [[[r.sub.1], [r.sub.2]].sup.n]] [member of] Z(R), for any [r.sub.1], [r.sub.2] [member of] I.
It is proved in [4, Proposition 4.2.6] that I is the largest liminal two-sided ideal of A.
As an example, let A be a two-sided ideal in a [C.sup.*]-algebra B .
Let ([[alpha].sub.n]) [member of] l and I be a two-sided ideal in a topological algebra A.
In particular, in all that follows we will make use of the previous cited result in the case I is a two-sided ideal of R.
A regular left (right) ideal of A is a left (respectively, right) ideal I of A for which there exists an element a [member of] A such that xa--x [member of] I (respectively, ax--x [member of] I) for all x [member of] A and a two-sided ideal I of A is regular if ax--x [member of] I and xa - x [member of] I for all x [member of] A.
It is well-known that, in a [GAMMA]-semigroup S, every hesitant fuzzy two-sided ideal is a hesitant fuzzy interior ideal of S, but the converse is not true in general.