Visual Group Theory, Lecture 7.3: Ring homomorphisms

Visual Group Theory, Lecture 7.3: Ring homomorphisms A ring homomorphism is a structure preserving map between rings, which means that f(x+y)=f(x)+f(y) and f(xy)=f(x)f(y) both must hold. The kernel is always a two-sided ideal. There are four isomorphism theorems for rings, which are completely analogous to the isomorphism theorems for groups. After stating these and several proof sketchs, we turn our attention to two different types of ideals: maximal and prime. A (proper) ideal M is maximal if no other proper ideal contains it. This is equivalent to the quotient ring R/M being a field. An ideal P is prime if xy being in P implies that either x or y is. This is equivalent to the quotient ring R/P being an integral domain. Course webpage (with lecture notes, HW, etc.): http://www.math.clemson.edu/~macaule/...