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Q: (B) Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.
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Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (An B) + (AnC).
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Q: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I= In rad R ???
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- Prove that if R is a field, then R has no nontrivial ideals.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.Label each of the following statements as either true or false. The only ideal of a ring R that property contains a maximal ideal is the ideal R.
- Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
- Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal ofTrue or false Label each of the following statements as either true or false. 3. The only ideal of a ring that contains the unity is the ring itself.21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.