a. Let
b. If
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Elements Of Modern Algebra
- 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 .)arrow_forward36. 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 .arrow_forwardLet 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.arrow_forward
- Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forwardLet R be a commutative ring that does not have a unity. For a fixed aR, prove that the set (a)={na+ra|n,rR} is an ideal of R that contains the element a. (This ideal is called the principal ideal of R that is generated by a. )arrow_forwardLet R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and only if r=r and s=s, (r1,s1)+(r2,s2)=(r1+r2,s1+s2), (r1,s1)(r2,s2)=(r1r2,s1s2). Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of R and S and is denoted by RS. Prove that RS is commutative if both R and S are commutative. Prove RS has a unity element if both R and S have unity elements. Given as example of rings R and S such that RS does not have a unity element.arrow_forward
- If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.arrow_forward37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forward21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,