If $R$ is a rng, show that $R\times \mathbb{Z}$ contains a subset in one to one correspondence with $R$.

Question

Let $(R,+,\cdot)$ be a rng (satisfies all the axioms of a ring except multiplicative identity). Define addition and multiplication in $R\times\mathbb{Z}$ by: $(a,n)+(b,m)=(a+b,n+m)$ and $(a,n)\cdot(b,m)=(ab+ma+nb,nm)$. Show that $(R\times \mathbb{Z},+,\cdot)$ is a ring that contains a subset in one-to-one correspondence with $R$ that has all the properties of the algebraic object $(R,+,\cdot)$.

Answer 1

This process adjoins an identity to a ring that may or may not have an identity, called a rng or a pseudo-ring. See if you can check that $(0,1)$ is a multiplicative identity in the ring $R\times \mathbb{Z}$ under the operation you've stated, and also that:

$$\varphi:R\hookrightarrow R\times \mathbb{Z}\\r\longmapsto (r,0)$$

is an injection of pseudo-rings. This means that the map respects addition and multiplication. I'll check addition, and leave multiplication for you:

$$\varphi(r+s)=(r+s,0)=(r,0)+(s,0)=\varphi(r)+\varphi(s)\qquad\checkmark$$