The Two Ideals of Fields

Summary

The Two Ideals of Fields By Susam Pal on 27 May 2025 A field has exactly two ideals: the zero ideal, which contains only the additive identity, and the whole field itself. These are known as trivial ideals. Further if a commutative ring, with distinct additive and multiplicative identities, has no ideals other than the trivial ones, then it must be a field. These two facts are elegant in their symmetry and simplicity. In this article, we will explore why these facts are true. Familiarity with algebraic structures such as groups, rings, and fields is assumed. Contents Definition of Ideals A left ideal of a ring \( R \) is a subset \( I \subseteq R \) such that \( I \) is an additive subgroup of \( R, \) and for all \( a \in I \) and \( r \in R, \) we have \( r \cdot a \in I. \) We say that a left ideal absorbs multiplication from the left by any ring element, or equivalently, that it is closed under left multiplication by any ring element. Similarly, a right ideal of a ring \( R \) is a subset \( I \subseteq R \) such that \( I \) is an additive subgroup of \( R, \) and for all \( a \in I \) and \( r \in R, \) we have \( a \cdot r \in I. \) We say that a right ideal absorbs multiplication from the right by any ring element, or equivalently, that it is closed under right multiplication by any ring element. In a commutative ring \( R, \) every left ideal is also a right ideal, and vice versa. This is because for all \( a \in I \) and \( r \in R, \) we have \( r \cdot a = a \cdot r. \) Therefore, when working with commutative rings, we do not need to distinguish between left and right ideals and we simply refer to them as ideals. In this case, the ideal is said to absorb multiplication by any ring element, or equivalently, it is said to be closed under multiplication by any ring element. Examples of Ideals Consider the set of even integers \[ \langle 2 \rangle = \{ 2n : n \in \mathbb{Z} \}. \] This is an ideal of \( \mathbb{Z}. \) Indeed, if we multiply any even integer...