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.

Q: Show, the set J=T+kery is the ideal ring P and y(Y))=g(p())=J .

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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: Show that in the factor ring Z[x] /(2x+1), the element x+(2x+1) is a unit.

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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: Let M2(Z) be a ring and =(( ), a,b e Z}. Then S is ideal %D .of M2(Z) l choose one true Error

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Q: 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.

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Q: Let R be a commutative ring. Prove that HO.R (R, M) and M are isomorphic R-modules.

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Q: Q2) Let (M₂ (R), +..) be a ring. Prove H = {(a) la, b, c €. = {(ab)la, b, c € R}is a subring of (M₂…

A: Subring Test : A non empty subset S of a ring R is a subring if S is closed under subtraction and…

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: 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: 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).

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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: 9. Suppose that (R,+, .) be a commutative ring with identity and x E rad R, then ..... .... (a) (x)…

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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: Is it true that if S is a unital subring of a unital ring, then the identity elements of the two…

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Q: 4 If R is a commutativ ring, show that the characteristic of R[X] is the same the characteristic of…

A: Given

Q: 9. Suppose that (R,+, .) be a commutative ring with identity and xE rad R, then (a) (x) = R (b) 1-x…

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

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

Q: Let R and S be commutative rings. Prove that (a, b) is a zero-divisorin R ⨁ S if and only if a or b…

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

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Q: If R is a commutative ring with unity and a e R, then (a) = {ar:reR}=aR. %3D %3D

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

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Q: 35. Show that the first ring is not isomorphic to the second. (a) Eand Z © Z × Zu and Z () ZXZ, and…

A: The objective is to show that the first ring of the following is not isomorphic to the second:

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: If Ris a ring with identity and a is a unit, prove that the equation ax = has a unique solution in…

A: Let R be a ring with identity and a∈R be a unit. Prove that the equation ax=b has a unique solution…

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

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Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for all…

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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: Suppose I, J be ideals of a commutative ring R. Prove that IJ cIn).

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Q: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

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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…

Q: Let R be a ring and M be an R-module. Ther

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Q: Theorem 12. Let R is a commutative ring and r e R,fe Hom (M, M'), then rfe. R M, M') defined by (rf)…

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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: Suppose that R and S are isomorphic rings. Prove that R[r] = S[r].

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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…

Q: If Ø: R → S is a ring isomorphism. The Ø preserves: Units Idempotent elements

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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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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.