the ring obtained from integer ring adjoining element Here is the problem from Algebra ,Artin : Determine the structure of the ring R' obtained from Z by adjoining element α satisfying relation 2α＝6
I've known what its elements is like. But   I don't know how to analyze the structure of the ring obtained by adjoining an element which satisfies a nonmonic polynomial relation and is not the inverse of any element of the original ring. I want a sample to show me how to analyze such a problem. Thankyou.
 A: Generally speaking, in an integral domain you can cancel nonzero elements even if they are not invertible: If $a\ne 0$ then $a\cdot b = a\cdot c \implies a(b-c)=0 \implies b=c$.
Adding algebraic elements to an integral domain gives you a subring of the algebraic closure of the field of fractions of the domain, and a subring of a field is always an integral domain, so $\mathbb{Z}[\alpha]$ is an integral domain, and you can apply this to get that $\alpha = 3$, i.e. $\mathbb{Z}[\alpha]=\mathbb{Z}$ (and the same can be said whenever you adjoint a root of a monic linear polynomial, because you again find out that you've "adjoined" an element already in the ring).
A: The first answer is valid under a natural but not explicitly stated assumption that both $\mathbb Z$ and $a$ are contained in a field or an integral domain. 
However, without this assumption $\mathbb Z$ can be considered as a subring of $\mathbb Z\oplus\mathbb Z_2$ (identifying $n$ with $(n,0)$) and for $a=(0,1)$ we have $\mathbb Z[a]=\mathbb Z\oplus\mathbb Z_2$. 
So the problem as stated does not have a unique solution.  
