Concept explainers
Prove the following statements for arbitrary elements
a. If
b. If
c. If
d. If
e. If
f. If
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Elements Of Modern Algebra
- 9. If denotes the unity element in an integral domain prove that for all .arrow_forwardComplete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .arrow_forwardIf x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xyarrow_forward
- If e is the unity in an integral domain D, prove that (e)a=a for all aD. [Type here][Type here]arrow_forward6. Prove that if is any element of an ordered integral domain then there exists an element such that . (Thus has no greatest element, and no finite integral domain can be an ordered integral domain.)arrow_forwardProve that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.arrow_forward
- [Type here] 21. Prove that ifand are integral domains, then the direct sum is not an integral domain. [Type here]arrow_forwardFor an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.arrow_forwardLet x and y be in Z, not both zero, then x2+y2Z+.arrow_forward
- 15. (See Exercise .) If and with and in , prove that if and only if in . 14. a. If is an ordered integral domain, prove that each element in the quotient field of can be written in the form with in . b. If with in , prove that if and only if in .arrow_forwardProve that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]arrow_forward14. a. If is an ordered integral domain, prove that each element in the quotient field of can be written in the form with in . b. If with in , prove that if and only if in .arrow_forward
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