# ring and field

• Mar 5th 2011, 05:50 AM
jin_nzzang
ring and field
how do we directly prove that if a ring is a field if and only if (0) is a maximal ideal? but without using the theorem that I is maximal if and only if R/I is a field

can anyone help me please ??
• Mar 5th 2011, 06:00 AM
tonio
Quote:

Originally Posted by jin_nzzang
how do we directly prove that if a ring is a field if and only if (0) is a maximal ideal? but without using the theorem that I is maximal if and only if R/I is a field

can anyone help me please ??

A unitary ring R is a field iff all its non-zero elements are invertible iff the principal ideal

$\langle r\rangle\,,\forall\,0\neq r\in R$ is the whole ring iff its (unique) proper ideal {0} is maximal.

Tonio