Hom worke: Consider the ring (Z [√3], +,.), Let A={ a+b√3:a, be Z₂} Is A subring of Z[√3]?

Q: Find the inverse of X + (X² – 8X + 7) in the form aX + b in the ring Q[X]/(X² – 8X + 7). - a b =…

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Q: Complete the proof of Theorem (the ring of cosests) by proving the right distributive law in R/I.

A: Let I be an ideal of the ring R. Then the set RI of additive cosets r+I of I in R forms a ring with…

Q: Q1. a) Give an example of a semi-ring that is not a semi -algebra. b) Give an example of a…

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Q: aw- Let Z[/2] = {a +b/2 |a, b e Z} and let H = { b а 2b : a,b eZ }. a Show that Z[ /2] and H are…

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Q: Show that the ring R={x + y√√2; x,y is under the set of real numbers} is a field

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Q: (a) Let R be a ring. State the definition of a zero-divisor of R. (b) Find the zero-divisors of the…

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Q: Define principal ideal ring.

A: Ideal: A subring A of a ring R is called a (two-sided) ideal of A if for every r∈R and every a∈A…

Q: (8) Show that for a field F, the set S of all all matrices of the form a b (88) 00 for a, b F is a…

A: An ideal is a special kind of subring of a ring R. If a∈I and r∈R implies that ra∈I then, the…

Q: Give an example of a ring that has exactly two maximal ideals

A: To Do: Give an example of a ring that has exactly two maximal ideals.

Q: R= 7 Z. Show that the polynomial (4,0)x in R[x] has infinitely meny roots in R.

A: Given : R = ℤ ⊕ ℤ To show that the polynomial (1, 0)x in R[x] has infinitely many roots in R.

Q: 5. Consider the ring Q[/2] (with the natural addition and multiplication). Is it an integral domain?…

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Q: Exercise 36. Let A,B, and C be ideals of a commutative ring R. Prove the following: (1) A.(B.C)…

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Q: 8 (R, +..) is an integral domain, then the ring (R x R, +..) is: (a) an integral domain (c) a simple…

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Q: 2- Let f be an isomorphism from the ring (R, +,) to the ring (R', +','). If (I, +;) is an ideal of…

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Q: 4. Prove that a zero divisor in a ring cannot be a unit.

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Q: How many distinct cosets are there in the factor ring Z[i] / ? Explain your answer.

A: Given: Let's determine the number of distinct cosets of the factor ring ℤi3+i.

Q: Given a ring R with elements {0, 1, a, b}, where: a+a=b, a×a=b, and b+b=0. Find the result of…

A: In abstract algebra, a ring is a set equipped with two binary operations, addition and…

Q: Let a ER which expressions are part of the Borel o algebra, B(R) they are correct? and give the…

A: The main objective is to determine the correct option.

Q: 2. Determine whether or not the set is an ideal in the ring M2×2(R). J-{(8₂9) | PCR} = ER

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Q: Consider a ring ZI/2)={a + b[/2]|a,bEZ}where addition and multiplication on ZI/21is defined as (a +…

A: A ring is commutative if all the elements of the ring commute under the second operation. Clearly,…

Q: Define the following principal ideals of the polynomial ring R[x]: I := (x³ + 1), J := (x + 1) and K…

A: Suppose  that   be  a  polynomial ring and  I = and .Then, IJ=Here, given that   be the  polynomial…

Q: Let x, y, z be elements of the real numbers. Use the field axioms and ordered axioms to prove -0=0.

A: We make use of the following field axioms:

Q: Which one of the following is not a commutative ring? (a) (Q. +, :) (b) (Z5, O5, 85) (c) (Z12, O12,…

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Q: The key to this particular problem is making sure all of the elements are of the form a + b5 Let (a…

A: Let a, b, c and d are integers then for ( a + b( 5^ 1/3 ) ) and ( c + d( 5^ 1/3 ) ) in A we have to…

Q: Q) Let (H,+10, -10) be a subring of the ring (Z10, +10. -10 ), where H = {[0], [2], [4], [6], [8]}.…

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Q: Prove that 2/3Z Z, as rings,

A: We shall be using First Isomorphism theorem of rings to solve this problem.

Q: Let R be a ring with no zero divisors, such that x^3 = x for each x in R(Real Number). Prove that…

A: Let R be a ring with no zero divisors  such that  We  have , We  have to  prove that  Characteristic…

Q: 3. Let f(x) = x³ + x³ + 2x² – 1 € Zg[x]. Then x = 1 is a root for f(x) of multiplicity: а) 1 b) 2 с)…

A: f(x)=x5+x3+2x2-1∈ℤ3 then f(x) has at x=1 (x-1) is factor of f(x)