Ring Theory/Ring extensions

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