Polynomial Rings and Ring Constructions

Polynomial rings, quotient rings, and localization are the fundamental construction tools for building new rings from old ones.

Polynomial Rings

Definition4.5Polynomial ring

For a commutative ring $R$ , the polynomial ring $R[x]$ consists of all finite formal sums $a_0 + a_1 x + \cdots + a_n x^n$ with $a_i \in R$ , with addition and multiplication defined in the standard way. More generally, $R[x_1, \ldots, x_n] = R[x_1]\cdots[x_n]$ is the ring of polynomials in $n$ variables.

The ring $R[x]$ satisfies a universal property: for any ring homomorphism $\varphi: R \to S$ and element $s \in S$ , there exists a unique ring homomorphism $\tilde{\varphi}: R[x] \to S$ with $\tilde{\varphi}|_R = \varphi$ and $\tilde{\varphi}(x) = s$ (evaluation at $s$ ).

Theorem4.3Hilbert basis theorem

If $R$ is a Noetherian ring (every ideal is finitely generated), then $R[x]$ is also Noetherian. Consequently, $R[x_1, \ldots, x_n]$ is Noetherian for any $n$ .

Quotient Rings

Definition4.6Quotient ring

For an ideal $I \subseteq R$ , the quotient ring $R/I$ consists of cosets $r + I$ with operations $(r + I) + (s + I) = (r+s) + I$ and $(r+I)(s+I) = rs + I$ . The projection $\pi: R \to R/I$ is a surjective ring homomorphism with $\ker(\pi) = I$ .

ExampleImportant quotient rings

$\mathbb{Z}/(n) = \mathbb{Z}/n\mathbb{Z}$ : integers modulo $n$ .
$\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}$ : the complex numbers arise as a quotient.
$k[x,y]/(y^2 - x^3 + x) \cong k[\bar{x}, \bar{y}]$ where $\bar{y}^2 = \bar{x}^3 - \bar{x}$ : the coordinate ring of an elliptic curve.
$k[x]/(x^n) = k[\varepsilon]$ with $\varepsilon^n = 0$ : a ring with nilpotent elements (the "dual numbers" when $n = 2$ ).

The Isomorphism Theorems for Rings

Theorem4.4Ring isomorphism theorems

First isomorphism theorem: If $\varphi: R \to S$ is a ring homomorphism, then $R/\ker(\varphi) \cong \mathrm{im}(\varphi)$ .
Second isomorphism theorem: If $I, J$ are ideals of $R$ with $I \subseteq J$ , then $(R/I)/(J/I) \cong R/J$ .
Chinese remainder theorem: If $I_1, \ldots, I_n$ are pairwise comaximal ideals ( $I_j + I_k = R$ for $j \neq k$ ), then $R/(I_1 \cap \cdots \cap I_n) \cong R/I_1 \times \cdots \times R/I_n$ .

ExampleChinese remainder theorem

$\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ since $(2) + (3) = \mathbb{Z}$ .

More generally, $\mathbb{Z}/n\mathbb{Z} \cong \prod \mathbb{Z}/p_i^{e_i}\mathbb{Z}$ for $n = \prod p_i^{e_i}$ .

RemarkLocalization

For a multiplicative set $S \subseteq R$ (closed under multiplication, $1 \in S$ , $0 \notin S$ ), the localization $S^{-1}R$ consists of fractions $r/s$ with $r \in R$ , $s \in S$ , modulo the equivalence $r/s \sim r'/s'$ iff $t(rs' - r's) = 0$ for some $t \in S$ . Important special cases: $R_\mathfrak{p} = (R \setminus \mathfrak{p})^{-1}R$ (localization at a prime), $R_f = \{1, f, f^2, \ldots\}^{-1}R$ .