§ 1. Algebras

1. Definition of an algebra

The $A$-bilinear mapping of $E \times E$ into $E$ occurring in this definition is called the multiplication of the algebra $E$; it is usually denoted by $(x, y) \mapsto x.y$, or simply $(x, y) \mapsto xy$.

Let $(\alpha_i)_{i \in I}$ and $(\beta_j)_{j \in J}$ be two families of elements of $A$, of finite support (I, § 2, no. 1). Then, for all families $(x_i)_{i \in I}$ and $(y_j)_{j \in J}$ of elements of $E$, the general distributivity formula (I, § 3, no. 4)

holds; in particular

The bilinear mapping $(x, y) \mapsto yx$ of $E \times E$ into $E$ and the $A$-module structure on $E$ define an $A$-algebra structure on $E$, called opposite to the given algebra structure. The set $E$ with this new structure is called the opposite algebra to the algebra $E$; it is often denoted by $E^0$. The $A$-algebra $E$ is called commutative if it is identical with its opposite, in other words if multiplication in $E$ is commutative. An isomorphism of $E$ onto $E^0$ is also called an anti-automorphism of the algebra $E$.

When multiplication in the algebra $E$ is associative, $E$ is called an associative $A$-algebra. When multiplication in $E$ admits an identity element (necessarily unique (I, § 2, no. 1)), this element is called the unit element of $E$ and $E$ is called a unital algebra.

Clearly the composition of two A-algebra homomorphisms is an A-algebra homomorphism. Every bijective algebra homomorphism is an isomorphism. Therefore A-algebra homomorphisms may be taken as morphisms of the species of A-algebra structure (Set Theory, IV, 2, no. 1). We shall always suppose in what follows that this choice of morphisms has been made. If E, E' are two A-algebras, let $\operatorname{Hom}_{A\text{-alg}}(E, E')$ denote the set of A-algebra homomorphisms of E into E'.

Let E, E' be two algebras each with a unit element. A homomorphism of E into E' mapping the unit element of E to the unit element of E' is called a unital homomorphism (or unital algebra morphism).

2. Subalgebras. Ideals. Quotient algebras

Let A be a commutative ring and E an A-algebra. If F is a sub-A-module of W which is stable under the multiplication on E, the restriction to F × F of the multiplication of E defines (with the A-module structure on F) an A-algebra structure on F. F, with this structure, is called a subalgebra of the A-algebra E. Every intersection of subalgebras of E is a subalgebra of E. For every family $(x_i)_{i \in I}$ of elements of E, the intersection of the subalgebras of E containing all the $x_i$ is called the subalgebra of E generated by the family $(x_i)_{i \in I}$ and $(x_i)_{i \in I}$ is called a generating system (or generating family) of this subalgebra. If $u : E \to E'$ is an A-algebra homomorphism, the image $u(F)$ of every subalgebra F of E is a subalgebra of E'.

Let E be an associative algebra. For every subset M of E, the set $M'$ of elements of E which are permutable with all the elements of M is a subalgebra of E called the centralizer subalgebra of M in E (I, § 1, no. 5). The centralizer $M''$ of $M'$ in E is also called the bicentralizer of M; clearly $M \subset M''$. It follows that $M'$ is contained in its bicentralizer $M''$, which is just the centralizer of $M''$; but the relation $M \subset M''$ implies $M'' \subset M'$, so that $M' = M''$ (cf. Set Theory, III, § 1, no. 7, Proposition 2). If F is a subalgebra of E, the centre of F is the intersection $F \cap F'$ of F and its centralizer $F'$ in E. Note that if

F is commutative, then $F \subset F'$ and hence $F' \supset F''$; the bicentralizer $F''$ of F is in this case the centre of F.

For certain non-associative algebras (for example Lie algebras) the notions of centralizer of a subalgebra and centre are defined differently (Lie Groups and Lie algebras, I, § 1, no. 6).

A subset $a$ of an A-algebra E is called a left ideal (resp. right ideal) of E when $a$ is a sub-A-module of E and the relations $x \in a,\ y \in E$ imply $yx \in a$ (resp. $xy \in a$). It amounts to the same to say that $a$ is a left ideal of E or a right ideal of the opposite algebra $E^0$. A two-sided ideal of E is a subset $a$ of E which is both a left ideal and a right ideal. When E is associative and admits a unit element $e$, then, for $\alpha \in A$ and $x \in E$, $\alpha x = (\alpha e)x = x(\alpha e)$ by virtue of (2) (no. 1) and hence the (right, left, two-sided) ideals of the ring E (I, § 8, no. 6) are identical with the (right, left, two-sided) ideals of the algebra E. Every sum and every intersection of left (resp. right, two-sided) ideals of the algebra E is a left (resp. right, two-sided) ideal. The intersection of the left (resp. right, resp. two-sided) ideals containing a subset X of E is called the left (resp. right, resp. two-sided) ideal of E generated by X.

Let $b$ be a two-sided ideal of an A-algebra E. If $x \equiv x' \pmod{b}$ and $y \equiv y' \pmod{b}$, then

Let E, $E'$ be two A-algebras and $u : E \to E'$ an algebra homomorphism. The image $u(E)$ is a subalgebra of $E'$ and the kernel $b = u^{-1}(0)$ is a two-sided ideal of E; further, in the canonical decomposition of $u$:

Let A be a commutative ring and E an A-algebra. On the set $\tilde{E} = A \times E$, we define the following laws of composition:

It is immediately verified that $\tilde{E}$, with these laws of composition, is an algebra over $A$ and $(1,0)$ is a unit element of this algebra. The set $\{0\} \times E$ is a two-sided ideal of $\tilde{E}$ and $x \mapsto (0,x)$ is an isomorphism of the algebra $E$ onto the subalgebra $\{0\} \times E$, by means of which $E$ and $\{0\} \times E$ are identified. $\tilde{E}$ is called the algebra derived from $E$ by adjoining a unit element; it is associative (resp. commutative) if and only if $E$ is.

3. Diagrams expressing associativity and commutativity

Let $A$ be a commutative ring and $E$ an $A$-module; being given a bilinear mapping of $E \times E$ into $E$ is equivalent to being given an $A$-linear mapping:

(II, § 3, no. 5). An $A$-algebra structure on $E$ is therefore defined by giving an $A$-module structure on $E$ and an $A$-linear mapping of $E \otimes_A E$ into $E$.

Let $E'$ be another $A$-algebra and $m' : E' \otimes_A E' \to E'$ the $A$-linear mapping defining the multiplication of $E'$. A mapping $f : E \to E'$ is an $A$-algebra homomorphism if and only if $f$ is a mapping rendering commutative the diagram

For an $A$-algebra $E$ to be associative, it is necessary and sufficient (taking account of the associativity of tensor products, cf. II, § 3, no. 8) that the diagram of $A$-linear mappings

be commutative. Similarly, for the $A$-algebra $E$ to be commutative, it is necessary and sufficient that the diagram of $A$-linear mappings

be commutative, where $\sigma$ denotes the canonical $A$-linear mapping defined by $\sigma(x \otimes y) = y \otimes x$ for $x \in E, y \in E$ (II, § 3, no. 1, Corollary 2 to Proposition 1).

For all $c \in E$, let $\eta_c$ denote the A-linear mapping of A into E defined by the condition $\eta_c(1) = c$. For $c$ to be a unit element of E, it is necessary and sufficient that the two diagrams

be commutative ($i$ and $i'$ denoting the canonical isomorphisms (II, § 3, no. 4, Proposition 4)).

Let E be an A-algebra with unit element e and let $\eta = \eta_e$ (also denoted by $\eta_E$); then $\eta(\alpha \beta) = \eta(\alpha) \eta(\beta) = \alpha \eta(\beta)$, for by (2) (no. 1),

hence $\eta$ is an A-algebra homomorphism. Observe that the A-module structure on E can be defined using $\eta$, for

(where, on the right hand side, multiplication is in E). The image of the homomorphism $\eta$ is a subalgebra of E whose elements commute with all those of E. The kernel of the homomorphism $\eta$ is the annihilator of the element e of the A-module E; by (3), it is also the annihilator of the A-module E (II, § 1, no. 12).

When the algebra E is unital and associative, $\eta$ is a ring homomorphism. Conversely, let $\rho : A \to B$ be a ring homomorphism such that the image $\rho(A)$ is contained in the centre of B, assuming also that the ring A is commutative; then an A-algebra structure is defined on B which is associative and unital, by writing (cf. (3))

4. Products of algebras

Let $(E_i)_{i \in I}$ be a family of algebras over the same commutative ring A. It is immediately verified that on the product set $E = \prod_{i \in I} E_i$, the product A-module structure (II, § 1, no. 5) and the multiplication

define an A-algebra structure; with this structure, the set E is called the product algebra of the family of algebras $(E_i)_{i \in I}$.

When all the algebras $E_i$ are associative (resp. commutative, resp. unital), so is their product. Moreover, all the properties stated in I, § 8, no. 10, extend without modification to arbitrary products of algebras.

5. Restriction and extension of scalars

Let $A_0$ and A be two commutative rings and $\rho : A_0 \to A$ a ring homomorphism. If E is an A-algebra, we denote (conforming with II, § 1, no. 13) by $\rho_* (E)$ the $A_0$-module defined by addition on E and the external law

Multiplication in E and the $A_0$-module structure on $\rho_* (E)$ define an $A_0$-algebra structure on $\rho_* (E)$. When $A_0$ is a subring of A and $\rho$ the canonical injection, the algebra $\rho_* (E)$ is said to be obtained from E by restricting the ring A of scalars to $A_0$. By an abuse of language, this is also sometimes said when $\rho$ is arbitrary.

Let F be an $A_0$-algebra. An $A_0$-algebra homomorphism $F \to \rho_* (E)$ is called a semi-homomorphism (relative to $\rho$) or a $\rho$-homomorphism of F into the A-algebra E; it is also called an $A_0$-homomorphism if no confusion arises. If E, $E'$ are two A-algebras, every A-algebra homomorphism $E \to E'$ is also an $A_0$-algebra homomorphism $\rho_* (E) \to \rho_* (E')$.

Consider now two commutative rings A and B and a ring homomorphism $\rho : A \to B$. For every A-module E, the B-module $\rho^*(E) = E \otimes_A B$, obtained from E by extending the ring A of scalars to B, has been defined (II, § 5, no. 1). If E is also an A-algebra, we shall define on $\rho^*(E)$ a B-algebra structure. For this, observe that $(E \otimes_A B) \otimes_B (E \otimes_A B)$ is canonically isomorphic to $(E \otimes_A E) \otimes_A B$ (II, § 5, no. 1, Proposition 3). If $m : E \otimes_A E \to E$ defines the multiplication on E, the mapping $m \otimes 1_B : (E \otimes_A E) \otimes_A B \to E \otimes_A B$ is therefore canonically identified with a B-linear mapping

which defines the desired B-algebra structure on $\rho^*(E)$. Hence

(5)

for $x, x'$ in E, $\beta$ and $\beta'$ in B. The B-algebra $\rho^*(E)$ is said to be derived from the A-algebra E by extending the ring A of scalars to B (by means of $\rho$). It is also denoted by $E_{(B)}$ or $E \otimes_A B$. When E is associative (resp. commutative, resp. unital), so is $\rho^*(E)$.

6. Inverse and direct limits of algebras

Let $I$ be a preordered set and $(A_i, \phi_{ij})$ an inverse system of commutative rings with $I$ as indexing set. Let $(E_i, f_{ij})$ be an inverse system of $A_i$-modules with $I$ as indexing set (II, § 6, no. 1) and suppose further that each $E_i$ has an $A_i$-algebra structure and that, for $i \leq j$, $f_{ij}$ is an $A_j$-homomorphism of algebras (relative to $\phi_{ij}$) (no. 5). Let $A = \lim \leftarrow A_i$ and $E = \lim \leftarrow E_i$, which has an $A$-module structure, the inverse limit of the structure of the $A_i$-modules $E_i$ (II, § 6, no. 1); it is immediately verified that the law of composition on $E$, considered as the inverse limit of the $E_i$ considered as magmas under multiplication (I, § 10, no. 1), with the $A$-module structure on $E$, defines on $E$ an $A$-algebra structure; $(E_i, f_{ij})$ is called an inverse system of $A_i$-algebras and the $A$-algebra $E$ is called its inverse limit. If $f_i : E \to E_i$, $\phi_i : A \to A_i$ are the canonical mappings, $f_i$ is an $A$-homomorphism of algebras (relative to $\phi_i$). If the $E_i$ are associative (resp. commutative), so is $E$; if each $E_i$ admits a unit element $e_i$ and $f_{ij}(e_j) = e_i$ for $i \leq j$, $e = (e_i)$ is a unit element of the algebra $E$.

Let $(E'_i, f'_{ij})$ be another inverse system of $A_i$-algebras and for all $i$ let $u_i : E_i \to E'_i$ be an $A_i$-algebra homomorphism, these mappings forming an inverse system; then $u = \lim u_i$ is an $A$-algebra homomorphism.

Suppose now that all the $A_i$ are equal to the same commutative ring $A$ and the $\phi_{ij}$ to $\mathrm{Id}_A$, so that $E = \lim \leftarrow E_i$ is an $A$-algebra. Let $F$ be an $A$-algebra and, for all $i \in I$, let $u_i : F \to E_i$ be an $A$-algebra homomorphism such that $(u_i)$ is an inverse system of mappings; then $u = \lim \leftarrow u_i$ is a homomorphism of the algebra $F$ into the algebra $E$. Conversely, for every $A$-algebra homomorphism $v : F \to E$, the family of $v_i = f_i \circ v$ is an inverse system of $A$-algebra homomorphisms such that $v = \lim \leftarrow v_i$. As moreover, writing $\bar{f}_{ij} = \mathrm{Hom}(1_F, f_{ij})$, $(\mathrm{Hom}_{A\text{-alg.}}(F, E_i), \bar{f}_{ij})$ is clearly an inverse system of sets, it is seen that the above remarks can also be expressed by saying that the canonical mapping $v \mapsto (f_i \circ v)$ is a bijection

Moreover, for every $A$-algebra homomorphism $w : F \to F'$, the

form an inverse system of mappings and the diagram

is commutative.

Suppose now that $I$ is right directed. Consider a direct system of commutative rings $(A_i, \phi_{ji})$ and a direct system $(E_i, f_{ji})$ of $A_i$-modules, with $I$ as indexing set; suppose that each $E_i$ has an $A_i$-algebra structure and that, for $i \leq j$, $f_{ji}$ is an $A_i$-homomorphism of algebras (relative to $\phi_{ji}$) (no. 5). Let $A = \lim \to A_i$, $E = \lim \to E_i$; $E$ has an $A$-module structure, the direct limit of the structures of the $A_i$-modules $E_i$ (II, § 6, no. 2); moreover, the law of composition on E considered as the direct limit of the E_i, considered as magmas under multiplication (I, § 10, no. 3), with the A-module structure on E, defines an A-algebra structure on E; (E_i, f_{ji}) is called a direct system of A_i-algebras and the A-algebra E is called its direct limit. If f_i : E_i \to E, \phi_i : A_i \to A are the canonical mappings, f_i is an A_i-homomorphism of algebras (relative to \phi_i). If the E_i are associative (resp. commutative), so is E; if each E_i admits a unit element e_i and f_{ji}(e_i) = e_j for i \leq j, E admits a unit element e such that f_i(e_i) = e for all i \in I.

Let (E'_i, f'_{ji}) be another direct system of A_i-algebras and for all i let u_i : E_i \to E'_i be an A_i-algebra homomorphism, these mappings forming a direct system; then u = \lim \to u_i is an A-algebra homomorphism.

Suppose now that all the rings A_i are equal to the same ring A and the \phi_{ji} to Id_A, so that E = \lim \to E_i is an A-algebra. Let F be an A-algebra and for all i let u_i : E_i \to F an A-algebra homomorphism such that (u_i) is a direct system of mappings; then u = \lim \to u_i is a homomorphism of the algebra E into the algebra F. Conversely, for every A-algebra homomorphism v : E \to F, the family of v_i = v \circ f_i is a direct system of A-algebra homomorphisms such that v = \lim \to v_i. As moreover, writing \vec{f}_{ij} = \mathrm{Hom}(f_{ij}, 1_F), (\mathrm{Hom}_{A\text{-alg.}}(E_i, F), \vec{f}_{ij}) is clearly an inverse system of sets, it is seen that the above remarks can also be expressed by saying that the canonical mapping v \mapsto (v \circ f_i) is a bijection

Further, for every A-algebra homomorphism w : F \to F', the

form an inverse system of mappings and the diagram

is commutative.

7. Bases of an algebra. Multiplication table

By definition, a basis of an A-algebra E is a basis of E for its A-module structure. Let (a_i)_{i \in I} be a basis of E; there exists a unique family (\gamma^k_{ij})_{(i, j, k) \in I \times I \times I} of elements of the ring A such that for every ordered pair (i, j) \in I \times I, the set of k \in I such that \gamma^j_{ij} \neq 0 is finite and

The $\gamma_{ij}^k$ are called the constants of structure of the algebra E with respect to the basis $(a_i)$ and relations (7) constitute the multiplication table of the algebra E (relative to the basis $(a_i)$).

Relations (7) can be imagined written down by setting out the right hand sides of these relations in a square table

it being understood that the element appearing in the row of index $i$ and the column of index $j$ is equal to the product $a_i a_j$.

Conversely, given an A-module E, a basis $(a_i)_{i \in I}$ of E and a family $(\gamma_{ij}^k)$ of elements of A such that for every ordered pair $(i, j) \in I \times I$ the set of $k \in I$ such that $\gamma_{ij}^k \neq 0$ is finite, then there is on E one and only one A-algebra structure under which relations (7) hold, since the A-module $E \otimes_A E$ is free and admits as basis $(a_i \otimes a_j)_{(i, j) \in I \times I}$ (cf. II, § 3, no. 7, Corollary 2 to Proposition 7).

Let E be an A-algebra and $(a_i)_{i \in I}$ a generating system of the A-module E (for example a basis). For E to be associative, it is necessary and sufficient that the $a_i$ satisfy the associativity relations

The mapping $(x, y, z) \mapsto (xy)z - x(yz)$ is an A-trilinear mapping

Similarly, for E to be commutative, it is necessary and sufficient that the $a_i$ satisfy the commutativity relations

The proof is analogous, this time considering the A-bilinear mapping $(x, y) \mapsto xy - yx$. Finally, for an element $e \in E$ to be a unit element, it is necessary and sufficient that the $a_i$ satisfy the relations

as is seen this time by considering the A-linear mappings $x \mapsto x - xe$ and $x \mapsto x - ex$.

When $(a_i)_{i \in I}$ is a basis of E and $(\gamma_{ij}^k)$ the corresponding family of constants of structure, relations (8) are equivalent to the relations

for all $i, j, k, s$. Similarly relations (9) are equivalent to $\gamma_{ij}^k = \gamma_{ji}^k$ for all $i, j, k$.

Let $(a_i)_{i \in I}$ be a basis of the A-algebra E; if $\rho : A \to B$ is a ring homomorphism, $(a_i \otimes 1)$ is a basis of the B-algebra $\rho^*(E) = E \otimes_A B$ (II, § 5, no. 1, Proposition 4). If $(\gamma_{ij}^k)$ is the family of constants of structure relative to the basis $(a_i)$, the family $(\rho(\gamma_{ij}^k))$ is the family of constants of structure of $\rho^*(E)$ relative to the basis $(a_i \otimes 1)$.

Exercises

See the exercises for § 1.