Since all of the noncommutative symmetric functions in this section commute and satisfy the same defining relations as their commutative counterparts, there is a

ring isomorphismIf V=eA then f eAe[right arrow][E.sub.A](eA) is a

ring isomorphism. Thus in particular [E.sub.A]([A.sub.A])[approximately equal to]A[approximately equal to][E.sub.A]([sub.A]A).

is a

ring isomorphism onto a subring of [Cart.sub.p]R ([14; IV, 4]).

Suppose B [[subset].bar] A and S [[subset].bar] R are ring extensions with

ring isomorphism [PSI] : A [??] R restricting to a

ring isomorphism [eta] : B [??] S.

Moreover, if [phi] is a bijection, then [phi] is called a neutrosophic

ring isomorphism and we write [R.sub.1](I) [congruent to] [R.sub.2](I).

mapping as R([j.sub.L]), is a

ring isomorphism such that R([j.sub.L]) = i x [[phi].sub.L], for the identical embedding i: R x L [right arrow] RL.

(1) There exist a

ring isomorphism [PHI] : W([K.sub.1]) [right arrow] W([K.sub.2]) and a group isomorphism [PSI] : [W.sub.q]([K.sub.1]) [right arrow] [W.sub.q]([K.sub.2]) such that [PSI](b.q) = [PHI](b).[PSI](q) for all b [member of] W([K.sub.1]) and for all q [member of] [W.sub.q]([K.sub.1]).

It follows that {[[bar.x].sup.l][[bar.y].sup.m]|0 [??] l [??] n - 1, 0 [??] m [??] 2n - 2} is a Z-basis of Z[x,y]/I Consequently, [bar.[??]] is a Z-module isomorphism, and so it is a

ring isomorphism.

Since the

ring isomorphism [Z.sub.n] [congruent to] [Z.sub.p] x [Z.sub.q], the projectors [mu], [eta] from Y to U = [Z.sub.p] and V = [Z.sub.q] satisfy the hypothesis in Proposition 4.1.

Under the

ring isomorphism that fixes the elements in [mathematical expression not reproducible] and [mathematical expression not reproducible] is isomorphic to the ring [mathematical expression not reproducible], where [u.sup.2] = 0.

Then (f, g) is called soft neutrosophic

ring isomorphism.

the upper horizontal map is strict, and an isomorphism, corresponding to the

ring isomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with (x, y) sent to (x'y, y) in k[x', y].