Dependencies

Let \(R\) be a commutative ring. An algebra \(A\) over \(R\) is a pair \((A, \bullet )\), satisfying:

1. \(A\) is a ring under \(*\).

2. there exists a ring homomorphism from \(R\) to \(A\), denoted \(\mathit{1}: R \to _{+*} A\).

3. \(\bullet : R \to M \to M\) is a scalar multiplication

4. for every \(r \in R\), \(x \in A\), we have

   1. \(r * x = x * r\) (commutativity between \(R\) and \(A\))

   2. \(r \bullet x = r * x\) (definition of scalar multiplication)

where we omitted that the ring homomorphism \(\mathit{1}\) is applied to \(r\) before each multiplication.

Let \(A\) and \(B\) be \(R\)-algebras. \(\mathit{1}_A\) and \(\mathit{1}_B\) are ring homomorphisms from \(R\) to \(A\) and \(B\), respectively.

A algebra homomorphism from \(A\) to \(B\) is a map \(f : \alpha \to _{a}\beta \) such that

1. \(f\) is a ring homomorphism

2. \(f(\mathit{1}_{A}(r)) = \mathit{1}_{B}(r)\) for each \(r \in R\)

Let \(R\) be a ring, \(M\) an \(R\)-module. An bilinear form \(B\) over \(M\) is a map \(B : M \to M \to R\), satisfying:

1. \( B(x + y, z) = B(x, z) +B(y, z) \)

2. \( B(x, y + z) = B(x, y) +B(x, z) \)

3. \( B(a \bullet x, y) = a * B(x, y)\)

4. \( B(x, a \bullet y) = a * B(x, y)\)

for all \(a \in R, x, y, z \in M\).

A division ring is a ring \((R, +, *)\), satisfying:

1. \(R\) contains at least 2 elements.

2. for all \(a \neq 0\) in \(R\), there exists a multiplicative inverse \(a^{-1} \in R\) such that

   \[ a * a^{-1} = a^{-1} * a = 1 \]

The dual module \(M^* : M \to _{l[R]} R\) is the \(R\)-module of all linear maps from \(M\) to \(R\).

Let \(X\) be an arbitrary set. An free \(R\)-algebra \(A\) on \(X\) (or “ generated by \(X\) ") is the ring quotient of the following inductively constructed set \(A_{\mathtt{pre}}\)

1. for all \(x\) in \(X\), there exists a map \(X \to A_{\mathtt{pre}}\).

2. for all \(r\) in \(R\), there exists a map \(R \to A_{\mathtt{pre}}\).

3. for all \(a, b\) in \(A_{\mathtt{pre}}\), \(a + b\) is in \(A_{\mathtt{pre}}\).

4. for all \(a, b\) in \(A_{\mathtt{pre}}\), \(a * b\) is in \(A_{\mathtt{pre}}\).

by that equivalence relation that makes \(A\) an \(R\)-algebra, namely:

1. there exists a ring homomorphism from \(R\) to \(A_{\mathtt{pre}}\), denoted \(R \to _{+*} A_{\mathtt{pre}}\).

2. \(A\) is a commutative group under \(+\).

3. \(A\) is a monoid under \(*\).

4. left and right distributivity under \(*\) over \(+\).

5. \(0 * a \sim a * 0 \sim 0\).

6. for all \(a, b, c\) in \(A\), if \(a \sim b\), we have

   1. \(a + c \sim b + c\)

   2. \(c + a \sim c + b\)

   3. \(a * c \sim b * c\)

   4. \(c * a \sim c * b\)

   (compatibility with the ring operations \(+\) and \(*\))

A group is a pair \((G, *)\), satisfying:

1. \((a * b) * c = a * (b * c)\) for all \(a, b, c \in G\) (associativity).

2. there exists \(1 \in G\) such that \(1 * a = a * 1 = a\) for all \(a \in G\).

3. for each \(a \in G\) there exists \(a^{-1} \in G\) such that \(a * a^{-1} = a^{-1} * a = 1\).

Let \(R, S\) be rings, \(M\) an \(R\)-module, \(N\) an \(S\)-module. A linear map from \(M\) to \(N\) is a function \(f : M \to _{l} N\) over a ring homomorphism \(\sigma : R \to _{+*} S\), satisfying:

1. \(f(x + y) = f(x) + f(y)\) for all \(x, y \in M\).

2. \(f(r \bullet x) = \sigma (r) \bullet f(x)\) for all \(r \in R\), \(x \in M\).

Let \(R\) be a commutative ring. A module over \(R\) (in short \(R\)-module) is a pair \((M, \bullet )\), satisfying:

1. M is a group under \(+\).

2. \(\bullet : R \to M \to M\) is called scalar multiplication, for every \(a, b \in R\), \(x, y \in M\), we have

   1. \(a \bullet (x + y) = a \bullet x + b \bullet y\)

   2. \((a + b) \bullet x = a \bullet x + b \bullet x\)

   3. \(a * (b \bullet x)=(a * b) \bullet x\)

   4. \(1_R \bullet x = x\)

A monoid is a pair \((R, *)\), satisfying:

1. \((a * b) * c = a * (b * c)\) for all \(a, b, c \in R\) (associativity).

2. there exists an element \(1 \in R\) such that \(1 * a = a * 1 = a\) for all \(a \in R\)

   i.e. \(1\) is the multiplicative identity (neutral element).

Let \(R\) be a commutative ring, \(M\) a \(R\)-module. An quadratic form \(Q\) over \(M\) is a map \(Q : M \to R\), satisfying:

1. \( Q(a \bullet x) = a * a * Q(x)\) for all \(a \in R, x \in M\).

2. there exists a companion bilinear form \(B : M \to M \to R\), such that \(Q(x + y) = Q(x) + Q(y) + B(x, y)\)

In some literatures, the bilinear form is denoted \(\Phi \), and called the polar form associated with the quadratic form \(Q\), or simply the polar form of \(Q\).

A ring is a triple \((R, +, *)\), satisfying:

1. \(R\) is a commutative group under \(+\).

2. \(R\) is a monoid under \(*\).

3. for all \(a, b, c \in R\), we have

   1. \(a * (b + c) = a * b + a * c\)

   2. \((a + b) * c = a * c + b * c\)

   (left and right distributivity over \(+\)).

Let \((\alpha , +_\alpha , *_\alpha )\) and \((\beta , +_\beta , *_\beta )\) be rings.

A ring homomorphism from \(\alpha \) to \(\beta \) is a map \(\mathit{1}: \alpha \to _{+*}\beta \) such that

1. \(\mathit{1}(x +_{\alpha } y) = \mathit{1}(x) +_{\beta } \mathit{1}(y)\) for each \(x,y \in \alpha \).

2. \(\mathit{1}(x *_{\alpha } y) = \mathit{1}(x) *_{\beta } \mathit{1}(y)\) for each \(x,y \in \alpha \).

3. \(\mathit{1}(1_{\alpha }) = 1_{\beta }\).

Let \(R\) be a non-commutative ring, \(r\) an arbitrary equivalence relation on \(R\). The ring quotient of \(R\) by \(r\) is the quotient of \(R\) by the strengthen equivalence relation of \(r\) such that for all \(a, b, c\) in \(R\):

1. \(a + c \sim b + c\) if \(a \sim b\)

2. \(a * c \sim b * c\) if \(a \sim b\)

3. \(a * b \sim a * c\) if \(b \sim c\)

i.e. the equivalence relation is compatible with the ring operations \(+\) and \(*\).

Let \(A\) be a free \(R\)-algebra generated by module \(M\), let \(\iota : M \to A\) denote the map from \(M\) to \(A\).

An tensor algebra over \(M\) (or “of \(M\) ") \(T\) is the ring quotient of the free \(R\)-algebra generated by \(M\), by the equivalence relation satisfying:

1. for all \(a, b\) in \(M\), \(\iota (a + b) \sim \iota (a) + \iota (b)\).

2. for all \(r\) in \(R\), \(a\) in \(M\), \(\iota (r \bullet a) \sim r * \iota (a)\).

i.e. making the inclusion of \(M\) into an \(R\)-linear map.

If \(R\) is a division ring, then a module \(M\) over \(R\) is called a vector space.