Maximal, prime, and radical ideals via their quotient rings

Let \(R\) be a commutative ring, and let \(\mfm\), \(\mfp\), and \(J\) be ideals of \(R\).

• \(R/\mfm\) is a field if and only if \(\mfm\) is a maximal ideal.

• \(R/\mfp\) is an integral domain if and only if \(\mfp\) is a prime ideal.

• \(R/J\) is reduced if and only if \(J\) is radical.