Prime Ideals A prime ideal is an ideal that is irreducible, meaning it cannot be expressed as the direct sum of two smaller ideals. In other words, a prime...
Prime Ideals
A prime ideal is an ideal that is irreducible, meaning it cannot be expressed as the direct sum of two smaller ideals. In other words, a prime ideal is not the union of two smaller ideals.
For example, consider the ideal of all polynomials of degree 2 in the ring of real numbers. This ideal is prime because it cannot be expressed as the direct sum of two smaller ideals.
Maximal Ideals
A maximal ideal is an ideal that is maximal, meaning it is not contained in any other ideal. In other words, a maximal ideal is the largest ideal that is not contained in any other ideal.
For example, consider the ideal of all polynomials of degree 1 in the ring of real numbers. This ideal is maximal because it is not contained in any other ideal.
Relationship between Prime and Maximal Ideals
A prime ideal is a maximal ideal if and only if the ideal is irreducible. This means that the ideal is not the direct sum of two smaller ideals