2) Let P + Q be maximal ideals in a ring R and a,b elements of R. Show that there exists c E R, such that a – c E P and b – c E Q.

Q: Show that the union of a chain I1 ⊂ I2 ⊂ .... of ideals of a ring R isan ideal of R.

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Q: 7. Suppose that (R,+,.) be a commutative ring with identity and (I,+,.) be an ideal of R. If I is…

A: Given that (R, +, .) be a commutative ring with identity and (I, +, .)  be an ideal of R such that I…

Q: Show that A is ideal of A+B? IS an

A: According to questionWe have to show that A is an ideal of A+B

Q: Let S = {0, 2, 4, 6, 8} CZ10. Is S an ideal of Z10?

A:

Q: Q1. a) Give an example of a semi-ring that is not a semi -algebra. b) Give an example of a…

A:

Q: 1. Let I and J be ideals of a ring R. Prove that IJ is an ideal of R.

A:

Q: 2. Let B and C be a ideals of a ring R. Prove that the set B - C = {teR such that t = f -g for some…

A:

Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: (b) (-a)b = −(ab) = a(−b) for all a,b € R.

A:

Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+(AnC) = (A B) + (AnC).

A:

Q: 5. Let R be a ring and I, J ideals of R. Is IUJ always an ideal of R? Either prove that this is the…

A: The objective of the question is to determine whether the union of two ideals I and J of a ring R is…

Q: Show that the union of a chain I, CI,c of ideals of a ring R is an ideal of R. (This exercise is…

A: Given: We are given a chain of ideals in a ring Objective: The objective is to show that the union…

Q: = Let I andJ be two ideals of a commutative ring R. Show that: S= {r€R: ri E J Vi EI } is an ideal…

A: Ideal: A subset of a ring is called ideal of if  1. 2. Given: are ideals in a ring and . To…

Q: Show that an intersection of ideals of a ring R is again an ideal of R.

A:

Q: Q2: Prove that the intersection of any two ideals of a ring R is also an ideal.

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Q: 2. Determine whether or not the set is an ideal in the ring M2×2(R). J-{(8₂9) | PCR} = ER

A:

Q: Show that a commutative ring R with unity is a field if and only if its only two ideals are (0) and…

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Q: . If A, B and Ç are ideals of a ring R, prove that A (B+ C) = AB+AC.

A:

Q: '. Suppose that (R, +,.) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

A: R is ring with unity and I is ideal of R.

Q: Let R be a ring. Let p= ax + b and q = cx² + dx + e be elements of R[x], for a, b, c, d, e E R. True…

A:

Q: If S be an ideal ofg a ring Z of all integers. Then S is maximal if and only if it is generated by…

A: If S be an ideal ofg a ring Z of all integers. Then S is maximal if  and only if it is generated by…

Q: Let R be a commutative ring with unity and let N={ aER | a"=0 for nez*, n>1}. Show that N is an…

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Q: If U, V are ideals of a ring R, let U + V = {u+ v:u E U,v E V}. Prove that U +V is also an ideal.

A: We have to prove the conditions of ideal

Q: Suppose A and B are ideal of the ring R. if A C

A: I have used definition of ideal of ring.

Q: Let A be an ideal of a ring R. i) If R is commutative then show that R/A is commutative ii) If R…

A: