1.2  Rings

Definition 1.2.1. A ring is an ordered pair \((R,+,\times)\) where \(R\) is a set, \(+\) is a binary operation called addition, and \(\times\) is a binary operation called multiplication satisfying the following axioms:

  1. the ordered pair \((R,+)\) is an abelian group;
  2. multiplication is associative;
  3. multiplication is distributive with respect to addition, that is, for all \(x,y,z\in R\), the following hold:
    • \(x\times (y+z) = (x\times y) + (x\times z)\) (left-distributive);
    • \((x+y)\times z = (x\times z) + (y\times z)\) (right-distributive).

We shall denote multiplication by juxtaposition, the additive identity of \(R\) by \(0\), and the additive inverse of a ring element \(x\) by \(-x\).

The distributive laws express that multiplication is biadditive: left multiplication and right multiplication are both additive. More precisely, for each fixed \(a\in R\), the maps \(x\mapsto ax\) and \(x\mapsto xa\) are homomorphisms of the additive group \((R,+)\) to itself.

Definition 1.2.2. A ring \(R\) is said to be commutative if multiplication is commutative. A ring \(R\) is said to have a multiplicative identity if there is an element \(1\in R\) such that \(1x = x1 = x\) for all \(x\in R\).

Proposition 1.2.3. Let \(R\) be a ring.

  1. We have \(0x=x0=0\) for all \(x\in R\).
  2. We have \((-x)y=x(-y)=-(xy)\) for all \(x,y\in R\).
  3. We have \((-x)(-y) = xy\) for all \(x,y\in R\).
  4. If \(R\) has a multiplicative identity \(1\), then the identity is unique and \(-x = (-1)x\) for all \(x\in R\).

Proof. (1) Since \(x+0 = x\), right distributivity gives \(xy = 0y +xy\) for all \(y\in R\). Hence \(0y=0\) for all \(y\in R\). Similarly, left distributivity gives \(yx = y0 + yx\), so that \(y0=0\) for all \(y\in R\). (2) Since we have \(x+(-x)=0\), right distributivity gives \(xy + (-x)y = 0y\). Since \(0y=0\) by (1), \(xy + (-x)y = 0\). Therefore \((-x)y = -(xy)\). Similarly, left distributivity gives and (1) gives \(yx + y(-x) = 0\). Thus \(y(-x) = -(yx)\). (3) Since \((-x)(-y) = -(x(-y))\) and \(x(-y) = -(xy)\) by (2), we have \((-x)(-y) = -(-(xy))\). Since \(-(-x) = x\), \((-x)(-y) = xy\). (4) Let \(1\) and \(1'\) be multiplicative identities. Then \(1 = 1\times 1' = 1'\). Hence a multiplicative identity is unique. Since

\[x + (-1)x = 1x + (-1)x =(1+(-1))x=0x=0,\]

we have \((-1)x = -x\).\(\square\)

Definition 1.2.4. A ring \(R\) with multiplicative identity \(1\ne 0\) is called a division ring if every nonzero element \(x\in R\) has a multiplicative inverse, that is, there exists \(y\in R\) such that \(xy = yx = 1\). A commutative division ring is called a field.

Definition 1.2.5. A subring of a ring \(R\) is a subgroup of \(R\) that is closed under multiplication.

Proposition 1.2.6. A subset \(S\) of a ring \(R\) is a subring of \(R\) if and only if

  1. the subset \(S\) is nonempty;
  2. the subset \(S\) is closed under subtraction;
  3. the subset \(S\) is closed under multiplication.

Proof. Suppose that \(S\) is a subring of \(R\). By Proposition 1.1.11, \(S\) is nonempty and closed under subtraction. By Definition 1.2.5, \(S\) is closed under multiplication. We now prove the converse. Since \(S\) is nonempty and is closed under subtraction, \(S\) is a subgroup of \(R\) by Proposition 1.1.11. Since \(S\) is closed under multiplication, \(S\) is a subring of \(R\) by Definition 1.2.5.\(\square\)