# A problem about prime subring

This is a problem from gtm 167 Field and Galois Theory.

Let R be a commutative ring with identity.The prime subring of R is the intersection of all subrings of R.Show that this intersection is a subring of R that is contained inside all subrings of R.Moreover,show that the prime subring of R is equal to $$\{n\cdot1:n\in\mathbb{Z}\}$$,where 1 is the multiplicative identity of R

The first part of this problem is easy.What puzzled me is the second part.I think under this definition, the prime subring of R should be $$0$$.Even if we only consider the inersection of non-zero subring of R,I think the outcome won't be $$\{n\cdot1:n\in\mathbb{Z}\}$$.Since $$\{2n\cdot1:n\in\mathbb{Z}\}$$ is also a subring of R.This is a controdiction.I want to know where I was wrong

• Is $2\mathbb{Z}$ considered a subring of $\mathbb{Z}$? It does not contain the identity of $\mathbb{Z}$. – Lozenges May 31 '19 at 5:29
• @Lozenges It will make sense if we ask the subring of R should contain the identity of R. But I remenber the definition of subring don't require that... – Y.Wayne May 31 '19 at 9:23