Let
be arbitrary rings. In the Cartesian product
and
Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of
Prove that
Prove
has a unity element if both
have unity elements.
Given as example of rings
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Chapter 5 Solutions
Elements Of Modern Algebra
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forwardExercises If and are two ideals of the ring , prove that is an ideal of .arrow_forward24. 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_forward
- 14. Let be an ideal in a ring with unity . Prove that if then .arrow_forwardAssume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.arrow_forwardProve that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.arrow_forward
- 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 .arrow_forwardLet :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?arrow_forwardProve that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forward
- 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.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_forward14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,