Ring Theory

Chapter 7 - Introduction to Rings

Def. A ring \(R\) is a set together with two binary operations \(+\) and \(\times\), satisfying the following conditions:

• \((R, +)\) is an abelian group,
• \(\times\) is associative,
• the distributive laws hold in \(R\).

The ring \(R\) is commutative if \(\times\) is commutative.

The ring \(R\) is said to have an identity if there is an element \(1 \in R\) with \(1 \times a = a \times 1 = a\) for all \(a \in R\).

From now on we will use \(ab\) to refer to \(a \times b\).

Remark. 规定 \(+\) 需要交换的一个重要原因是保证乘法结果唯一，例如 \((1+1)(a+b)\)。

Def. A ring \(R\) with identity \(1 \ne 0\) is called a division ring if every nonzero element \(a \in R\) has a multiplicative inverse. A commutive division ring is called a field.

Exp.

• Trivial rings: Any commutative group and define \(ab = 0\) for \(\forall a, b \in R\).
• Zero ring: \(R = \{0\}\).
• The rings of \(\mathbb Z\), \(\mathbb Q\), \(\mathbb R\) and \(\mathbb C\).
• The (real) Hamilton Quaternions: Let \(H = \{a + bi + cj + dk\}\), where \(a, b, c, d \in \mathbb R\). It is noncommutative and its identity is \(1 = 1 + 0i + 0j + 0k\). The real (or rational) Hamilton Quaternions is a division ring.
• Rings of functions. Let \(X\) be any nonempty set and \(A\) be any ring. The collection \(R\) of all functions \(f: X \to A\) is a ring under the usual definition of pointwise addition and multiplication of functions.

Prop. 7.1. Let \(R\) be a ring. Then

• \(0a = a0 = 0\) for \(\forall a \in R\).
• \((-a)b = a(-b) = -ab\) for \(\forall a, b \in R\).
• \((-a)(-b) = ab\) for \(\forall a, b \in R\).
• if \(R\) has an identity \(1\), then the identity is unique and \(-a = (-1)a\) for \(\forall a \in R\).

Def. Let \(R\) be a ring.

• A nonzero element of \(R\) is called a zero divisor if there is a nonzero element \(b \in R\) s.t. \(ab = 0\) or \(ba = 0\).
• Assume \(R\) has an identity \(1 \ne 0\). An element \(u\) of \(R\) is called a unit in \(R\) if there is some \(v \in R\) s.t. \(uv = vu = 1\). The set of units in \(R\) is denoted as \(R^\times\).

Exp.

• Let \(n \ge 2\). \(u\) is a unit in \(\mathbb Z / n \mathbb Z\) iff \((u, n) = 1\), and otherwise \(u\) is a zero divisor.

Remark. unit 和 zero divisor 虽然无交，但是其并集不一定是全集，例如 \(\mathbb Z\) 中的 \(2\)。

Def. A commutative ring with identity \(1 \ne 0\) is called an integral domain if it has no zero divisors.

Prep. 7.2. Assume \(a, b, c \in R\), and \(a\) not a zero divisor. If \(ab=ac\) then either \(a=0\) or \(b=c\).

Proof. \(a(b-c) = 0\). If \(a \ne 0\), since \(a\) is not a zero divisor, \(b-c=0\).

Cor. 7.3. Any finite integral domain is a field.

Proof. For any \(a \ne 0\), \(ab \ne ac\) for all \(b \ne c\). Since the ring is finite, there must exist \(b\) s.t. \(ab = 1\).

Def. A subring of the ring \(R\) is a subgroup of \(R\) that is closed under multiplication.

Exp.

• \(\mathbb Z\) is a subring of \(\mathbb Q\), and \(\mathbb Q\) is a subring of \(\mathbb R\).
• Let \(D\) be a square-free integer, then \(\mathbb Z(D) = \{a+b\sqrt D \mid a, b \in \mathbb Z\}\) forms a subring of \(\mathbb Q(D) = \{a+b\sqrt D \mid a, b \in \mathbb Q\}\).

Def. polynomial ring

Exp. \(\mathbb Z[x]\), \(\mathbb Q[x]\), \(\mathbb Z/ 3\mathbb Z[x]\).

Remark. the ring in which the coefficients of the polynomials are taken makes a substantial difference. For example, \(x^2+1\) is not a perfect square in \(\mathbb Z[x]\), but it is in \(\mathbb Z / 2 \mathbb Z[x]\).

Prop. 7.4. Let \(R\) be an integral domain and let \(p(x), q(x)\) be nonzero elements of \(R[x]\), then

• \(\deg (pq) = \deg p + \deg q\).
• The unis of \(R[x]\) are precisely the units of \(R\).
• \(R[x]\) is an integral domain.

Def. matrix ring

Def. Fix a commutative ring \(R\) with identity \(1 \ne 0\) and let \(G = \{g_1, g_2, \cdots, g_n\}\) be any finite group with group operation written multiplicatively. The group ring \(RG\) is the set of all formal sums \(\sum_{g \in G} a_g g\), where \(a_g \in R\) for all \(g \in G\), and the addition and multiplication are defined by $$\left(\sum_{g \in G} a_g g\right) + \left(\sum_{g \in G} b_g g\right) = \sum_{g \in G} (a_g + b_g) g,$$ $$\left(\sum_{g \in G} a_g g\right)\left(\sum_{h \in G} b_h h\right) = \sum_{g, h \in G} a_g b_h (gh).$$

The ring \(RG\) is commutative iff \(G\) is abelian.

Def. Let \(R\) and \(S\) be rings.

• A ring homomorphism from \(R\) to \(S\) is a map \(\varphi: R \to S\) s.t. \(\varphi(a + b) = \varphi(a) + \varphi(b)\) and \(\varphi(ab) = \varphi(a)\varphi(b)\) for \(\forall a, b \in R\).
• The kernel of \(\varphi\) is the set \(\ker \varphi = \{a \in R \mid \varphi(a) = 0\}\).
  A bijective ring homomorphism is called a ring isomorphism.

Prop. 7.5. Let \(\varphi: R \to S\) be a ring homomorphism.

• The image of \(\varphi\) is a subring of \(S\).
• The kernel of \(\varphi\) is a subring of \(R\). Furthermore, if \(\alpha \in \ker \varphi\) then \(r \alpha, \alpha r \in \ker \varphi\) for all \(r \in R\).

Proof.

• Trivial.
• \(\varphi(\alpha r) = \varphi(\alpha) \varphi(r) = 0\) and similarly \(\varphi(r \alpha) = 0\).

Remark. \(\ker \varphi\) 对于乘法也具有一些性质。与商群类似，我们在定义商环 \(R/S\) 时，\(S\) 同样需要满足这些性质。

Def. Let \(R\) be a ring and \(I\) a subset of \(R\). Define

• \(rI = \{ra \mid a \in I\}\) and \(Ir = \{ar \mid a \in I\}\) for \(r \in R\).
• A subset \(I\) of \(R\) is called a left ideal if
  • \(I\) is a subgroup of \(R\), and
  • \(rI \subseteq I\) for all \(r \in R\).
• Similarly we define a right ideal.
• A subset \(I\) of \(R\) is called a ideal if it is both a left ideal and a right ideal.

Def. Let \(I\) be an ideal of \(R\). The (additive) quotient ring \(R/I\) is the set of all cosets \(r + I\) for \(r \in R\), with addition and multiplication defined by $$ (r + I) + (s + I) = (r + s) + I, \quad (r + I)(s + I) = (rs) + I.$$

Prop. 7.6. We can verify that the addition and multiplication are well-defined. Conversely, if \(I\) is any subset of \(R\) s.t. the addition and multiplication are well-defined, then \(I\) is an ideal of \(R\).

Thm. 7.7.

• (The first isomorphism theorem for rings) Let \(\varphi: R \to S\) be a ring homomorphism. Then \(\ker\varphi\) is an ideal of \(R\), the image of \(\varphi\) is a subring of \(S\), and \(R/\ker \varphi\) is isomorphic to the image of \(\varphi\).
• If \(I\) is any idael of \(R\), then the map \(\varphi: R \to R/I\) defined by \(\varphi(r) = r + I\) is a ring homomorphism with kernel \(I\). This map is called the natural projection.

Thm. 7.8. Let \(R\) be a ring.

• (The second isomorphism theorem for rings) Let \(A\) be a subring and \(B\) be an ideal of \(R\), then \(A + B = \{a + b \mid a \in A, b \in B\}\) is a subring of \(R\), and \(A \cap B\) is an ideal of \(A\). Furthermore, \((A + B) / B \cong A / (A \cap B)\).
• (The third isomorphism theorem for rings) Let \(I\) and \(J\) be ideals of \(R\) with \(I \subseteq J\). Then \(J / I\) is an ideal of \(R / I\), and \((R / I) / (J / I) \cong R / J\).
• (The fourth or lattice isomorphism theorem for rings) Let \(I\) be an ideal of \(R\). The correspondence \(A \leftrightarrow A/I\) is an inclusion preserving bijection between the set of subrings \(A\) of \(R\) that contain \(I\) and the set of subrings of \(R/I\). Furthermore, \(A\) is an ideal of \(R\) iff \(A/I\) is an ideal of \(R/I\).

Def. Let \(I, J\) be ideals of \(R\). The sum of \(I\) and \(J\) is $$I + J = {a + b \mid a \in I, b \in J}.$$ The product of \(I\) and \(J\) is $$IJ = \left{ \sum_{i=1}^n a_i b_i \mid n \in \mathbb N, a_i \in I, b_i \in J \right}.$$

We see that \(I+J\) is the smallest ideal containing both \(I\) and \(J\), and \(IJ\) is the ideal contained in \(I\cap J\).

太难写了，放弃了，还是看课件吧。