# Rings250C7. ShowwheRoN2(a +b12la, b E Qzt il la + bila, bE ZQ8. Shoimp9. ProvZ.is a10. Ver7Z3Z2Z5Z11. Pro12. Let9Z6Z4Z10Za is13. De18Z12Z8Z14. LeFigure 12.1 Partial subring lattice diagram of C.m15. ShIn the next several chapters, we will see that many of the fundamen-riingroup theory can be naturally extended to rings. In par-oftal conceptsticular, we will introduce ring homomorphisms and factor rings16. Sn17. SExercises18. LThere is no substitute for hard work.THOMAS ALVA EDISON, Life19.1. Give an example of a finite noncommutative ring. Give an exampleof an infinite noncommutative ring that does not have a unity2. The ring (0, 2, 4, 6, 8) under addition and multiplication modulo10 has a unity. Find it.3. Give an example of a subset of a ring that is a subgroup underaddition but nota subring.20.21.22.4. Show, by example, that for fixed nonzero elements a and b in aring, the equation axdoes this compare with groups?b can have more than one solution. How235. Prove Theorem 12.2.6. Find an integer n that shows that the rings Z, need not have the fol-lowing properties that the ring of integers has.a. a = a implies a = 0 or a = 1.b. ab 0 implies a = 0 or b = 0.c. ab ac and a 0 imply b c.Is the n you found prime?2526272224

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Step 1

It is known that the ring Zn is an integral domain if and only if n is prime.

Step 2

(b)

Given that ab = 0 implies a = 0 or b =...

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