Does this opetation and structure have a name?

Question

A is a commutative ring ,a is an ideal of A. then we can get a structure A/a called quotient ring by operation of quotient.

question is :if we have a ring A/a and a set a . How to get A?

This operation we use like a inverse quotient. How to call the operation that we use A/a and a to get A.

Is there a name for A?

Answer 1

Think of the ring $\mathbb{Z}/ 4 \mathbb{Z}$ and the ring $(\mathbb{Z}/2 \mathbb{Z})^2$. Both give $\mathbb{Z}/ 2 \mathbb{Z}$ when you take the quotient by $\mathbb{Z}/ 2 \mathbb{Z}$. Thus, in general, $A$ is not uniquely defined.