is a Z-module isomorphism, and so it is a ring isomorphism
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.
From (8) and the ring isomorphism discussed above, we have
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].
In Section 2 we will see that there is a ring isomorphism
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 isomorphism
bar] R are ring extensions with ring isomorphism
[PSI] : A [?
Moreover, if [phi] is a bijection, then [phi] is called a neutrosophic ring isomorphism
and we write [R.
As [PHI] induces a factor ring isomorphism
u from [I.