Let I and J be ideals of a ring R.Prove that I+J= , the ideal of R generated by I ∪ J.
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Let I and J be ideals of a ring R.Prove that I+J= <I∪J>, the ideal of R generated by I ∪ J.
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- 15. Let and be elements of a ring. Prove that the equation has a unique solution.Exercises If and are two ideals of the ring , prove that the set is an ideal of that contains each of and . The ideal is called the sum of ideals of and .Let I1 and I2 be ideals of the ring R. Prove that the set I1I2=a1b1+a1b2+...+anbnaiI1,biI2,nZ+ is an ideal of R. The ideal I1I2 is called the product of ideals I1 and I2.
- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.