a. Show that the set
b. Is
c. does
d. Decide whether or not the set
Want to see the full answer?
Check out a sample textbook solutionChapter 6 Solutions
Elements Of Modern Algebra
- 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forwardGiven that the set S={[xy0z]|x,y,z} is a ring with respect to matrix addition and multiplication, show that I={[ab00]|a,b} is an ideal of S.arrow_forward15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forward
- 44. Consider the set of all matrices of the form, where and are real numbers, with the same rules for addition and multiplication as in. a. Show that is a ring that does not have a unity. b. Show that is not a commutative ring.arrow_forwardProve that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forward18. If possible, solve .arrow_forward
- Let [ a ] be an element of n that has a multiplicative inverse [ a ]1 in n. Prove that [ x ]=[ a ]1[ b ] is the unique solution in n to the equation [ a ][ x ]=[ b ].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
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning