A ring \(R\) is a set with binary operations \(+,\cdot\) such that

  1. \((A,+)\) is an abelian group

  2. \((A,\cdot)\) is a semigroup

  3. \(\cdot\) distributes over \(+\) on both sides

We only consider commutative rings with 1, that is, we require:

  1. \(x\cdot y = y\cdot x\) for all \(x, y \in R\)

  2. There exists \(1\in R\) with \(1\cdot x = x\cdot 1 = x\) for all \(x \in R\)

It is easily seen that if an identity element exists, it must be unique.

If \(1 = 0\) then we must have \(R = \{0\}\), which we call the zero ring.

A subset \(S\subset R\) is a subring if:

  1. \(1\in S\)

  2. \(x+y,x\cdot y,-x\in S\) for all \(x,y \in S\)

The latter condition is equivalent to requiring \(x-y,x\cdot y \in S\) for all \(x,y\in S\).

[In other contexts we do not require \(1\in S\), but simply that \(S\) is nonempty.]


  1. \(\mathbb{Z}\) is the only subring of \(\mathbb{Z}\).

  2. \(\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}\)

  3. The Gaussian integers \(\mathbb{Z}[i]\) form a subring of \(\mathbb{C}\).

  4. \(\mathbb{Z}_n\) is a ring. (Later we shall see this is the quotient ring \(\mathbb{Z}/n\mathbb{Z}\).)

  5. For any ring \(R\), let

    \(R[[x]] =\{a_0 + a_1 x + a_2 x^2 + ... | a_0,a_1,a_2... \in R\}\)

    This is the formal power series over \(R\). One subring of this is the ring of polynomials over \(R\):

    \( R[x] = \{a_0 + a_1 x + ... + a_n x^n | n \ge 0, a_0, ..., a_n \in R\} \)

Ben Lynn blynn@cs.stanford.edu 💡