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- [Type here] 18. Prove that only idempotent elements in an integral domain are and . [Type here]Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.[Type here] 21. Prove that ifand are integral domains, then the direct sum is not an integral domain. [Type here]
- Complete 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: .For 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.Let x and y be in Z, not both zero, then x2+y2Z+.
- Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both infinite?Prove the following problems in the space bellow.let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B belongs to t with x= A union B and A intersect B = zero