A ring isomorphic to a proper subring of itself

Question

Give an example of a ring which is isomorphic to a proper sub-ring of itself.

HINT: Consider $\Bbb R^\Bbb N$.

My try:As given in the hint I considered $\Bbb R^\Bbb N$ i.e the set of all sequences from $\Bbb N\rightarrow \Bbb R$ with pointwise multiplication and addition.

Consider the set $X=\{(x_n):x_0=0\}$ i.e. those sequences from $\Bbb R^\Bbb N$ whose first term is zero. This is a proper subset of $\Bbb R^\Bbb N$ and also a subring but the isomorphism given by

$a_0+a_1x+\ldots +a_nx^n+\ldots\mapsto a_1x+a_2x^2+\ldots a_nx^n\ldots$

is not working .How to find the correct one?

Answer 1

I would advice not to feel obliged to follow hints. The first example that comes to mind is the subring $\mathbf Z[X^2]$ of $\mathbf Z[X]$ of all polynomials in $X^2$ with integer coefficients. More precisely, let $$ e\colon \mathbf Z[X]\rightarrow \mathbf Z[X] $$ be the evaluation-at-$X^2$ morphism defined by $e(P)=P(X^2)$. It is injective, hence an isomorphism onto its image, the proper subring $\mathbf Z[X^2]$ of $\mathbf Z[X]$.

Answer 2

Consider the ring of polynomials in variables $x_1$, $x_2$, ... and its subring generated by the variables with even index.