# Ring Theory/Ring extensions

**Definition (ring extension)**:

Whenever is a ring and is a subring of , we say that is a **ring extension** of and write .

Note that if is a ring extension, then is a ring extension; indeed, the set is the set of all polynomials with coefficients in , the set is the set of all polynomials with coefficients in , and is a subring of .

**Proposition (existence of splitting ring)**:

Let be a ring, and let be a polynomial over . Then there exists a ring extension such that in , decomposes into linear factors, that is,

- for certain .

**Proof:** We prove the theorem by induction on the degree of . Suppose first that can be decomposed into two polynomials