а. Let a, b, c be three elements of a Euclidean ring R and (a, b) = 1. If a/bc then show that a/c.

Q: Let R be a euclidean ring and a,b are non zero elements of R.if a and b are associates,show that…

A: Let R be a Euclidean domain with measure d. Also given that a and b are associates in R. Then from…

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: (5) The ring (Z6, +6,•6 ) has that rad Z, (a) ((0), +6.-6 )

A: Given: To explain the correct option of the given statement as follows,

Q: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

A: Since, given that (M, + .) Is a maximal ideal of the commutative ring with identity (R, +, .)  and…

Q: Problem 3. (i) Prove or disprove: if R is a commutative ring and P and Q are polyno- mials in R[x],…

A: As per bartleby guidelines for more than one question asked only first should be answered. Please…

Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +, .) (c) ((0),…

A: The given question is solved with explanation below.

Q: Suppose that K is a commutative ring with identity. If and I are ideals of R for which R/I≈ R/J as…

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Q: If Ø: R S is a ring homomorphism. The Ø preserves:

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Q: The number of idempotents elements in the ring Zs is: O 2 4 8 1 O O O

A: Ans : 4  Option 2nd true

Q: Q:Let S and Szare two subrings of a ring (R, +..), prove that S, US2 subring of R iff either S, C S…

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Q: The number of zero divisors of the ring Z, Z5 is О 1 O 5

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

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Q: Let (R, +, ) be a ring, a, b e R with a + 0. Then the equation ax = b has a solution in R.

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Q: Let a and b be elements of a ring. Prove that (-a)b = -(ab).

A: Solve the following

Q: If A and B are ideals of a ring, show that the product of A and B, AB = {a,b, + a,b, +· · · + a,b,l…

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Q: Any ring R is a Jacobsen radical ring, if it is a simple ring. O مبا O

A: JACOBSON RADICAL:- The radical of the base ring R is called its Jacobson radical and denoted by…

Q: Show that the set 4Z U 5Z is not a subring of the integer ring Z:

A: Z = { ....., -3, -2, -1, 0, 1, 2, 3, ..... }. 4Z = { ...., -8, -4, 0, 4, 8, .... }. 5Z = { ....,…

Q: 5. An element x in a ring R is called idempotent if a2 = x. Prove that if a is an idempotent element…

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Q: Rp { -b a | a, b = Zp}

A: It is given that,  Rp = ab-ba : a, b ∈ ℤp We have to prove that,  Rp is a commutative ring with unit…

Q: If R is commutative ring then deg(f(x)g(x))= deg((S)) + deg(g(x)).

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Q: Let CR,t,,) be a Commutative ring ?

A: First I recall you Definition of commutative ring.  A commutative ring is ring in which…

Q: If R is commutative ring then deg(f(x) + g(x)) = max(deg(f(x)),deg(g(x)). %3D

A: We have to solve given problem:

Q: Determine all reversible elements in ring Z_6 and Z_8 and give their inverse

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Q: (4)If D is a ring of integers module 13. Then rad(D) = {0}.

A: given

Q: Find the charactes ist c of the ring Z2 Zg

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Q: Let R be Euclidean ring and a, b non - Zevo elements of R- If dca) <d lab), are show that b is not…

A: Let a and b be non-zero elements of R. Assume that b is a unit in R. Then, for some element c in R…

Q: Let R, , O is a ring under two composion e and O üs follows ü e i; = a + b + 1 and aOb = ab + a + b…

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Q: * If K is a simple ring, then K a Jacobsen radical ring. T F

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Q: Suppose that R is a commutative ring with unity 1. Then if ab is a zero divisor then a or b is a…

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Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +,.) (c) ((0),…

A: Let R be a ring. A two-sided ideal I of R is called maximal if I ≠ R and no proper ideal of R…

Q: If A and B are ideals of a commutative ring R with unity and A + B= R, show that A N B = AB.

A: Given Data: A and B are ideal of a commutative ring R with unity And A + B = R   For A⋂B = AB, The…

Q: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field. (b) If R is a…

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Q: The number of idempotents elements in the ring Zg is: 8 O 2 4 1

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Q: lisprove: that the triple <Z,,, constitutes a ring. et of units U(Z₂), and state whether (U(Z₁), Ⓒ)…

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Q: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…

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Q: be a Euclidean ring and asb elements of R- gf a and b Let R are non- ze YO asso ciates , Show that…

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Q: Show that if n is an integer and a is an element from a ring, thenn . (-a) = -(n . a).

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Q: An cxample of an infinite non-commutative ring with identity is: M,(Z) Ma (2z10) O This Option O…

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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: Prime Ideal:- An ideal P of a commutative ring R is prime if it has the following two properties:…

Q: 1- Every Boolcan algebra (B, V, A), becomes a Boolean ring (B, +, .) on defining addition and…

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Q: a. Let a, b, c be three elements of a Euclidean ring R and (a, b) = 1. If a/bc then show that a/c.

A: Here we use the definition of greatest common divisor.

Q: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

A:

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: 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…

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Q: The number of idempotents elements in the ring Zo is: 1 2 8 O 4

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Q: Consider the ring R = [r, s,t) whose addition and multiplications tables are given below, Then ts S…

A: Given R=r,s,t be the ring under addition and multiplication. Given table is +rstrrstsstrttrs We have…

Q: Show that the rings (3Z/60Z)/(12Z/60Z) and 3Z/12Z are isomorphic. Then show tha both isomorphic to…

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Q: P(X), where Is (P(X), An) is A= ring (A-B) ULB-A) a where

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Q: s{a+bV2:a, b e Z } under addition and multiplication a ring? Justify. Is it a mmutative ring?

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Q: Theorem 2.7. IfI, I, I2, . .., I, are ideals in a commutative ring R, then: (i) Rad (Rad I) Rad I;…

A: The solution for the above question is as shown below.

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.