5- The image of a commutative ring under homomorphism is commutative a) True b) False a) True O b) False
Q: 6- If fis a ring homomorphism from Z to Za such that f(1) = a, then a = a a) True b) False O a) True…
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Q: In commutative ring with identity each proper ideal is contained in prime ideal. O T OF
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Q: 6. Prove or disprove: the set of all subsets of R is a ring with respect to the operations A…
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Q: The image of a commutative ring under homomorphism may be noncommutative False True
A: As you asked multiple questions , I solved the first one. So you asked "the image of a commutative…
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A: Bb
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Q: Let (R, +,-) ring with properties X.XX, XER Cindempotent properties), show! @a = 0, x ER (b) x. y…
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Q: Consider the ring R = (0,2,4,6, 8, 10} under addition and multiplication modulo 12. The char (R) is
A: Characteristic of a Ring R: Char(R) is n. Where n is smallest number such that r+r+..+r(n…
Q: * (10 Let (R, +,) be commutative ring with identity. Then a - b = a-c-b=c ifand only if R has no…
A: True
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A: To prove that: (2, 5)=ℤ
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A: Option (b) is correct.
Q: A ring (R. +.) .) is commutative if addition is commutative in R. O True O False
A: Solve the following
Q: An example on a finite non- commutative ring without identity is:
A: Matrices are noncommutative So we can choose set of matrices Not if we take Mn[R] then this is non…
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Q: If R is a commutative ring and Ø:R→S is a ring homomorphism, then S is a commutative ring True O…
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Q: If a is an idempotent in a commutative ring, show that 1 - a is alsoan idempotent.
A: Here given
Q: 9. Suppose that (R,+, .) be a commutative ring with identity and x E rad R, then ..... .... (a) (x)…
A:
Q: Let o : Z, → Zn be a nontrívial ring homomorphism. (a)What can you say about n ?( Explain).…
A: See the detailed solution below.
Q: 2. Define reducible and irseducible elements ring R and irreducible elements in Zu, Zq nand Zio R…
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Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1 —…
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Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1- x…
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Q: Suppose that R is a commutative ring without zero-divisors. Showthat all the nonzero elements of R…
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A: Determine all ring homomorphisms from Q to Q.
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A: C will be right answer.
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Q: I is set f all integers (positic, is the ordinaiy multiplication negative and zero),. and show that…
A:
Q: 14) Any subring of a commutative ring is commutative. True False
A: I have given an answer in step 2.
Q: Suppose that R is a commutative ring without zero divisors . show that characterstic of R is 0 or…
A: suppose that R is a commutative ring without zero divisors to show- characteristic of Ring R is 0 or…
Q: Let a, b, and c be elements of a commutative ring, and suppose thata is a unit. Prove that b divides…
A: Let a, b, and c be elements of a commutative ring where a is a unit. Suppose that bdivides c. Then c…
Step by step
Solved in 3 steps with 3 images
- 21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.22. Let be a ring with finite number of elements. Show that the characteristic of divides .10. Prove or disprove that the set of all nonzero integers is closed with respect to a. addition defined on . b. multiplication defined on .
- 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.)15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .14. Letbe a commutative ring with unity in which the cancellation law for multiplication holds. That is, if are elements of , then and always imply. Prove that is an integral domain.
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.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 .46. Let be a set of elements containing the unity, that satisfy all of the conditions in Definition a, except condition: Addition is commutative. Prove that condition must also hold. Definition a Definition of a Ring Suppose is a set in which a relation of equality, denoted by , and operations of addition and multiplication, denoted by and , respectively, are defined. Then is a ring (with respect to these operations) if the following conditions are satisfied: 1. is closed under addition: and imply . 2. Addition in is associative: for all in. 3. contains an additive identity: for all . 4. contains an additive inverse: For in, there exists in such that . 5. Addition in is commutative: for all in . 6. is closed under multiplication: and imply . 7. Multiplication in is associative: for all in. 8. Two distributive laws hold in: and for all in . The notation will be used interchageably with to indicate multiplication.