The map a naturally induces a

ring homomorphism [??] : Z[[[pi].sub.1]([E.sub.K])] [right arrow] Z[[t.sup.[+ or -]1]], where Z[[[pi].sub.1]([E.sub.K])] is the group ring of [[pi].sub.1]([E.sub.K]).

The function [epsilon] : Z[[Q.sub.16]] [right arrow] Z given by [mathematical expression not reproducible] is a

ring homomorphism. Let Q be the field of rational numbers, [M.sub.2](Q) the ring of 2 x 2 matrices with entries in Q, Q[[square root of (2)]] = {a + b [[square root of (2)]] | a, b [member of] Q}, H(Q[[[square root of (2)]]]) the quaternion field and [Q.sup.4] [direct sum] [M.sub.2] (Q) [direct sum] H(Q[[square root of (2)]]) the ring with component-wise addition and multiplication.

Let f : R [right arrow] S be a

ring homomorphism. If [xi] is a 2-absorbing primary fuzzy ideal of S, then [f.sup.-1] ([xi]) is a 2absorbing primary fuzzy ideal of R.

Restricted to each stalk, a sheaf morphism O is assumed to be a

ring homomorphism. A morphism of ringed spaces is said to be (i) a monomorphism if [phi] is an injection and [THETA] is an epimorphism and (ii) an epimorphism if [phi] is a surjection, while [THETA] is a monomorphism.

Then [phi] is not a neutrosophic quadruple

ring homomorphism.

We define a

ring homomorphism [[theta].sub.[GAMMA]] on the ring of symmetric functions [lambda] by setting [[theta].sub.[GAMMA]] ([e.sub.0]) = 1 and

there is an injective

ring homomorphism [R.sub.1] [right arrow] R.

In any case f is a group homomorphism and if V=eA, f is a

ring homomorphism. If h[member of][Hom.sub.A](eA, V) then f(h(e))=h.

Definition 2.3: A mapping f: [R.sub.1] [right arrow] [R.sub.2] is said to be hemi

ring homomorphism of [R.sub.1] is to [R.sub.2] if f(x + y) = f(x) + f(y) and f(xy) = f(x).f(y) for all x, y [member of] R.

Throughout this section, let A be a unital associative ring and B [??] A a subring where [1.sub.B] = [I.sub.A]; more generally, it suffices to assume B [right arrow] A is a unital

ring homomorphism, called a ring extension, although we suppress this option notationally.

is a surjective

ring homomorphism. For each cyclic code C in [mathematical expression not reproducible] and i [member of] {0,1}, let