If R is a commutative ring with unity, show that every maximal ideal of R is a prime ideal.
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A: Since, given that (M, + .) Is a maximal ideal of the commutative ring with identity (R, +, .) and…
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A: Given: R is a ring with unity which has exactly one maximal left ideal M. To show: The only…
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Q: (B) Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.
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Q: Let R be a commutative ring with 10. Prove that R is a field if and only if 0 is a maximal ideal.
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A: To prove that every prime ideal of R is maximal.
Q: Prove that every field is a principal ideal ring.
A: We’ll answer the first part of this question since due to complexity. Please submit the question…
Q: 3) Given a commutative ring with unity 1 in R; where R is a ring with two maximal ideals M₁ and Show…
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Q: maximal
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Q: 20. Suppose R is a commutative ring and [R]= 30. If I is an ideal of R and |I| = 10, prove that I is…
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Q: show that an ideal A is maximal iff A is
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Q: Show that in a Boolean ring R, every prime ideal P is not equal to R is maximal.
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Q: Prove that in a ring R having exactly one maximal ideal M, the only idempotents are 0 and 1.
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Q: (b) If M is a maximal ideal of a ring R then M is a prime ideal of R.
A: Given: If M is a maximal ideal of a ring R, then M is a prime ideal of R To prove or disprove the…
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Q: Let R be a finite commutative ring with unity. Show that every prime ideal in R is a maximal ideal.
A: We have to show that if R be a finite commutative ring with unity then every prime ideal in R is a…
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A: Given: Ring P[x] and a subring which contains P.
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Q: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…
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Q: elements
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Q: Let R be a commutative ring with identity. Show that every maximal ideal is also a prime ideal. Give…
A: i just used the properties of maximal ideal to prove the result
Q: Let R be a commutative ring. Prove that HomR (R, M) and M are isomorphic R-modules.
A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…
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A: Image is attached with detailed solution.
Q: An element x in a ring is called an idempotent if x2 = x. Prove that the characterstic of R is 0 or…
A: An element x in a ring is called an idempotent if x^2 = x
Q: 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…
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Q: Find the maximal ideals and the prime ideals in the ring Z6.
A: We have to find the maximal ideals and the prime ideals in the ring Z6
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- 27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.22. Let be a ring with finite number of elements. Show that the characteristic of divides .36. Suppose that is a commutative ring with unity and that is an ideal of . Prove that the set of all such that for some positive integer is an ideal of .
- 21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.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 .)
- Let R be a commutative ring that does not have a unity. For a fixed aR, prove that the set (a)={na+ra|n,rR} is an ideal of R that contains the element a. (This ideal is called the principal ideal of R that is generated by a. )12. Let be a commutative ring with unity. If prove that is an ideal of.11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .