Rings

A ring is an abelian group \(R\) with binary operation \(+\) (“addition”), together with a second binary operation \(\cdot\) (“multiplication”). The operations satisfy the following axioms:

i) Multiplication is associative: For all \(a, b, c \in R\),

ii) The Distributive Law holds: For all \(a, b, c \in R\),

In other words, if we expand the definition of abelian group, then we can say that a non-empty set \(R\) is said to form a ring with respect to the binary operations addition \((+)\) and multiplication \((\cdot)\) provided, for arbitrary \(a\), \(b\), \(c\) \(\in R\), the following properties hold:

• Associative Law of addition

  \[(a+b)+c = a+(b+c)\]

• Commutative Law of addition

  \[a+b = b+a\]

• Existence of an additive identity (zero)

  There exists \(z \in R\) such that \(a+z = a\)

• Existence of additive inverses

  For each \(a \in R\), there exists \(-a \in R\) such that \(a+(-a)=z\)

• Associative Law of multiplication

  \[(a \cdot b) \cdot c = a \cdot (b \cdot c)\]

• Distributive Laws

  \[a(b+c) = a \cdot b + a \cdot c\]
• \[(b+c)a = b \cdot a + c \cdot a\] \[(b+c)a = b \cdot a + c \cdot a\]

The systems \(Z\), \(R\), \(Q\) and \(C\) are some examples of rings.

Types of Rings

Subrings

Let \(R\) be a ring. A non-empty subset \(S\) of the set \(R\), which is itself a ring with respect to the binary operations on \(R\), is called a subring of \(R\).

Homomorphism of Rings

Let \(R\) and \(S\) be rings. A ring homomorphism is a function \(f:R \rightarrow S\) such that:

a) For all \(x, y \in R\), \(f(x + y) = f(x) + f(y)\)
b) For all \(x, y \in R\), \(f(x y) = f(x)f(y)\)

Usually, we require that if \(R\) and \(S\) are rings with 1, then

c) \(f(1_R) = 1_S\)

The first two properties stipulate that \(f\) should “preserve” the ring structure - addition and multiplication.

Lemma 1: Let \(R\) and \(S\) be rings and let \(f: R \to S\) be a ring homomorphism.

a) \(f(0) = 0\) b) \(f(-r) = -f(r)\) for all \(r\in R\)

Lemma 2: Let \(R\), \(S\), and \(T\) be rings, and let \(f: R \to S\) and \(g: S \to T\) be ring homomorphisms. Then the composite \(g \cdot f: R \to T\) is a ring homomorphism.

Isomorphism of Rings

Let \(R\) and \(S\) be rings. A ring isomorphism from \(R\) to \(S\) is a bijective ring homomorphism \(f: R \to S\).
If there is a ring isomorphism \(f: R \to S\), \(R\) and \(S\) are isomorphic and this relationship is symbolically represented as \(R \approx S\).

For example, if \(R\) is a ring, the identity map \(\hbox{id}: R \rightarrow R\) is an isomorphism of \(R\) with itself.

Since a ring isomorphism is a bijection, isomorphic rings must have the same cardinality. So, for example, \(Z_6 \not\approx Z_{42}\), because the two rings have different numbers of elements.

However, \(Z\) and \(Q\) have the “same number” of elements (the same cardinality), but they are not isomorphic as rings. (Intuition: \(Z\) is a field, while \(Q\) is only an integral domain.)

Rings