Looking for a commutative ring satisfying certain conditions

Question

I'm looking for a commutative ring $R$ (with unit) which is of characteristic 2 and which possesses elements $x$ and $y$ such that the following holds

$x^2$ and $y^2$ are inverses of one another but $x$ and $y$ are not; $x, y, 1+x, 1+y$ are all units.

An example of such a ring or reasoning as to why one cannot exist would be very much appreciated.

Answer 1

We have $ (x^{-1})^2=(x^2)^{-1}=y^2$ and hence $$ (y-x^{-1})^2=0,$$ so $R$ has zero divisors. This motivates the following example:

Let $R=\mathbb F_4[\epsilon] =\mathbb F_4[X]/(X^2)$ and $x$ a generator of the cyclic group $F_4^\times$. Then $x^3=1$ and $x\ne 1$, so $x^2+x+1=0$. Let $y=x^2+\epsilon$. Then $y^2=x$. We verify

• $x\in R^\times$ because $x^3=1$
• $y\in R^\times$ because $y^2=x\in R^\times$
• $1+x\in R^\times$ because $1+x\in\mathbb F_4$ and $x\ne1$; or explicitly: $(1+x)^3=x+x^2=1$
• $1+y\in R^\times$ because $(1+y)^2=1+x\in R^\times$
• $xy=1+x\epsilon\ne1$
• $x^2y^2=x^2x=1$

It was not required, but in this example, $x, y, 1+x,1+y$ are even four distinct units.