Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division ring is a field.
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- Prove that a finite ring R with unity and no zero divisors is a division ring.24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.
- 27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.36. Suppose that is a commutative ring with unity and that is an ideal of . Prove that the set of all such that for some positive integer is an ideal of .
- If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.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.Suppose that a,b, and c are elements of a ring R such that ab=ac. Prove that is a has a multiplicative inverse, then b=c.