On Some Properties of Extensions of Commutative Unital Rings

Throughout the text, let R be a commutative ring with identity (often called a commutative unital ring) and with multiplicative group of units R∗. Likewise, let f(x) = a0x +a1x n−1+ · · ·+an−1x+an be a polynomial of the variable x over R such that a0 ∈ R∗. Traditionally, R[x] is the ring of all polynomials of x over R; thereby f(x) ∈ R[x]. For an arbitrary but fixed element α, suppose f(x) is the minimal polynomial in R[x] for which f(α) = 0, i. e. α is a root of f(x). Such an f(x) will be hereafter denoted by fα(x). Define R[α] = { r0 + r1α+ · · ·+ rn−1α| ri ∈ R, 0 6 i 6 n− 1 } .