Question

Let $I$ and $J$ be ideals of a ring $R$ such that $I+J=R$

(a) Prove that if $IJ=0$, then $R$ is isomorphic to the product ring $\dpi{100} (R/I)x(R/J)$

(b) Describe the idempotents corresponding to the product decomposition in (a) above.

(c) Show the ideals generated by each idempotent, and quotient that they correspond to in (b) above.

Please show all details so I may understand the process and compare the steps to my work. Thank you.

Answer 1

If IJ = 0, then R is isomorphic to the product ring R/J times R/I Soln Consider the map R rightarrow R/I times R/J that sends each x = a + b elementsof R R = I + J to (b, a) .This is a ring homomorphism with kernal I Intersection J. We can conclude with the first isomorphism theorems for rings. Theorem rightarrow (1) It phi: R rightarrow S is homomorphism of rings then the kernal of phi is an ideal of R, the image of phi is a subring of s and R/ker phi is isomorphic as a ring to phi(R) . (2) If I is any ideal of R, then the map R rightarrow R/I defined by r rightarrow r + I is a surjective ring homomorphism with kernal I. Thus every ideal is the kernal of a ring is homomorphism and vice versa. Let e elementsof I and e^' elementsof I be such that e + e^' = 1 we show that