1- The ring of integer numbers (Z+-) isa subring but not ideal of the ring of real numbers (R +) 2- Let Rand S be two rings with identity 1. 1s respectively. If fR-S isa ring homomorphism, then f(1,) - 1,.

Q: Prove that a commutative ring R that holds the cancellation law of multiplication has no zero…

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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…

Q: Let (R, +,-) ring with properties X.XX, XER Cindempotent properties), show! @a = 0, x ER (b) x. y…

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Q: ring R, an element a is a regular element if it is not a zero divisor in R. How many regular…

A: In general, an element a in a ring R is a zero divisor if there exists a nonzero element b in R such…

Q: Muslim Q2: (a) Let I be an ideal of the ring R, prove that f(I) is an ideal of R', when f is onto…

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Q: Let R be a ring with unity in which r^2 = r for every r in R. Prover that if a does not equal 0 and…

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Q: o which ring is ℤ_24/(3) isomorphic to? ℤ_2 ℤ_3 ℤ_4 ℤ_8 None of the above

A: To determine to which ring ℤ_24/(3) is isomorphic, we need to find a homomorphism φ : ℤ_24 → R such…

Q: Question 4: Mark each of the following by True (T) or False (F) 1) In a commutative ring with unity…

A: Given statements are ,  (1).   In a commutative ring with unity every unit is  a zero divisor . (2).…

Q: Theorem (3-6):- Let f:(R,+, .) → (R`, +' ,.) be a ring. homo , onto function and 1- If (R,+,.) is…

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Q: Solve the polynomial congruence: x³ - 3x² +27= 0 (mod 225).

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Q: There are. Polynomials of degree atmost n in the polynomial ring Z, [x]. none 5^(n+1) 5^n 5 + 5^n

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Q: Show that a homomorphism of rings whose domain is a field is either injective or the zero…

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Q: F Let K/L be ideal of quotient ring R/L. If K is prime ideal and contains L, then K/L is prime…

A: We can solve this using defination of ideals ,prime ideals ,quotient ideals

Q: Enter the multiplicative inverse of the element [6]18 X+ [5]18 in the ring Z18[x]. When you enter…

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Q: by example thai it is an sible to find ideals A,B,C R Such that a

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Q: Let Z be the ring of integers, and let R be a ring without identity. Let S = Z x R be the ring with…

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Q: There are . Polynomials of degree atmost n in the polynomial ring Zs (x]. O 5 + 5^n O 5^n O 5^(n+1)…

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Q: (b): Define reducible and irreducible elements in ring R and find all the reducible and irreducible…

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Q: 2. Let R be an integral domain (a) Define prime element and irreducible element (b) Prove every…

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Q: . Polynomials of degree atmost n in the polynomial ring Z,[x]. O 7+7^n O 7^(n+1) O none O 7^n

A: 72

Q: factor rings R/R and R/0 12. Give an example to show that a factor ring of an integral domain may be…

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Q: 24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent which is different…

A: R, +,· is said to be commutative ring if Suppose R is a non empty set such that for any two elements…

Q: Consider the following population table. 1970 1980 1990 2000 2010 2020 State A 11,110,485 11,427,509…

A: Population table is given.We find how much did the population of State C increase from 2000 to 2010.

Q: 7. The number of distinct homomorphisms from the ring (Z, +,.) onto the ring (Ze, +,.) is (a) 0 (b)…

A: Number of distinct onto homomorphism is

Q: A ring, R, in which r^2 = r for all elements r in R is called a Boolean ring. In any Boolean ring,…

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Q: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ........ O…

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Q: If fis a ring homomorphism from Zm to Z, such that f (1) = b, then b*+2 = b*. True False

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Q: (5) In commutative ring with identity each proper ideal is contained in prime ideal. От F

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Q: Poove or disaprove. that the Characterisiis 8 simple ung s either zero or is ing

A: A simple ring R is a ring (with unit) such that the only two-sided ideals are R and (0). The…

Q: Show that every ring can be embedded in a ring with identity.

A: We have to show that every ring can be embedded in a ring with identity.

Q: Suppose you want to prove that a ring homomorphism :R→S preserves nth powers of the ring, that is…

A: To prove: f(xk)=f(x)k