Abstract Algebra — Supplementary terminology

Supplementary terminology

These supplementary definitions were added for the linked reading edition. They are separate from the author’s original numbered statements.

Supplementary definition (Isomorphisms of algebras). §

Let $R$ be a commutative ring, and let $A$ and $B$ be $R$ -algebras. An $R$ -algebra isomorphism is a bijective $R$ -algebra homomorphism $A \to B$ . Thus it preserves addition, multiplication, the identity element, and the action of $R$ .

Supplementary definition (Embeddings of algebras). §

Let $R$ be a commutative ring, and let $A$ and $B$ be $R$ -algebras. An $R$ -algebra embedding is an injective $R$ -algebra homomorphism $A \to B$ . Neither algebra is required to be a field. In the remark on products of field embeddings in the number theory sample, $L \to L \otimes_{K} M$ is considered as an embedding of $K$ -algebras.

Supplementary definition (Finite generation as an algebra). §

Let $R$ be a commutative ring and let $B$ be a commutative $R$ -algebra. We say that $B$ is finitely generated as an $R$ -algebra if there are finitely many elements $b_{1}, \dots, b_{n} \in B$ such that every element of $B$ is a polynomial in the $b_{i}$ with coefficients from $R$ , acting through the given algebra structure. When $R$ is a subring of $B$ , this is written $B = R [b_{1}, \dots, b_{n}]$ . Algebra generation permits arbitrary products of the generators; module generation uses their linear combinations.

Supplementary definition (Square-free integers). §

An integer $d$ is square-free if $p^{2}$ does not divide $d$ for every prime number $p$ . Negative integers are allowed. This condition excludes $0$ .

Supplementary definition (Monomials in commuting elements). §

For commuting elements $u_{1}, \dots, u_{n}$ of a ring, a monomial in these elements is a product $u_{1}^{e_{1}} \dots u_{n}^{e_{n}}$ with nonnegative integer exponents $e_{1}, \dots, e_{n}$ . In particular, monomials in two commuting elements $α$ and $β$ have the form $α^{i} β^{j}$ with $i, j \geq 0$ .

Supplementary definition (Fixing elements and a base field). §

Let $L$ and $M$ be fields containing a field $K$ , and let $σ : L \to M$ be a field embedding. We say that $σ$ fixes an element $a \in L$ when $σ (a) = a$ . It fixes $K$ when it fixes every element of $K$ . An embedding that fixes $K$ therefore fixes the coefficients of every polynomial in $K [x]$ .

Supplementary definition (Leading terms of polynomials). §

If $f = a_{0} + a_{1} x + \dots + a_{n} x^{n}$ is a nonzero polynomial with $a_{n} \neq 0$ , its leading term is $a_{n} x^{n}$ . The element $a_{n}$ is its leading coefficient.

Supplementary definition (Direct products of algebras). §

Let $R$ be a commutative ring and let ${(A_{i})}_{i \in I}$ be a nonempty family of $R$ -algebras. Their direct product has the direct product of the underlying rings as its ring structure. Addition and multiplication are coordinate-wise, and the scalar action is $r \cdot {(a_{i})}_{i \in I} = {(r \cdot a_{i})}_{i \in I}$ . In the tensor-product lemma, every factor is $M$ , with one factor for each field embedding of $L$ into $M$ fixing $K$ .

Supplementary definition (Products of maps with a common domain). §

For a family of maps $f_{i} : X \to Y_{i}$ , the product of these maps is the map $X \to \prod_{i \in I} Y_{i}$ that sends $x$ to ${(f_{i} (x))}_{i \in I}$ . Thus every coordinate records the value of one of the maps at the same input. In the corresponding remark, the coordinate maps are the field embeddings $σ : L ↪ M$ that fix $K$ .

Supplementary definition (Finite sums and products in a ring). §

For elements $a_{1}, \dots, a_{n}$ of a ring with $n \geq 1$ , the sum $\sum_{i = 1}^{n} a_{i} = a_{1} + \dots + a_{n}$ uses repeated addition. The product $\prod_{i = 1}^{n} a_{i} = a_{1} \dots a_{n}$ uses repeated multiplication in the displayed order. Associativity makes a choice of parentheses unnecessary. A sum of products is formed by first multiplying finite lists of elements and then adding the resulting elements.

Supplementary definition (Positive integers). §

An integer $n$ is positive if $n > 0$ , or equivalently if $n \in {1, 2, 3, \dots}$ . In the tensor-product proposition, this describes the exponents $e_{i}$ in the polynomial factorization.

Supplementary definition (Odd integers). §

An integer $n$ is odd if $n = 2 k + 1$ for some integer $k$ . Equivalently, $n \equiv 1 (𝑚𝑜𝑑 2)$ . This includes negative odd integers.

Supplementary definition (Logical equivalence of conditions). §

Conditions $P_{1}, \dots, P_{n}$ are equivalent under given hypotheses if, whenever those hypotheses hold, each condition implies every other condition. For two conditions $P$ and $Q$ , this is expressed by saying $P$ holds if and only if $Q$ holds: both implications $P \Rightarrow Q$ and $Q \Rightarrow P$ are required.