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Q: ok but whats the answer im still confused
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A: The bionomial expansion of (1-x)-1 upto the term x2 is given below.
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Solved in 2 steps
- Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .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.
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R. (Hint: Compare this with Example 4, and note that the element a itself may not be in this set I.) b. Give an example of a commutative ring R and an element aR such that a(a)={ar|rR}.Let :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?
- 24. 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 .)32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].