Let Z be the ring of integers, and let R be a ring without identity. Let S = Z x R be the ring with addition and multiplication defined by (k, a) + (n, b) = (k + n, a + b) and (k, a)(n, b) = (kn, kb + na + ab), where k and n are in Z, and a and b are in R. Which of the following must be true about S? I. S is a ring with identity. II. S has a subring isomorphic to R. III. S is an integral domain (it has no zero- divisors). (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III 18

Q: D. Given the set F= {0, a, b, c, d} with operations + and * and the following tables, assume that…

A:

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: Show that the set Q(√2) = {a+b√2, a, b € Q} is a ring, under operations of multiplication and…

A: The set Q2=a+b2 , a , b∈Q  and operations under multiplication and addition define by :…

Q: Let R = {0, 2, 4, 6, 8}. Then R is a ring under addition and multiplication (mod 10). Find the…

A:

Q: Find the Fourier cosine integral representation of the following: f(t) = {t 0 < t < 1 elsewhere

A:

Q: Let us define two new operations on R called ~ and ∗ by the following formulas. For real numbers a…

A:

Q: ompute the greatest common divisor in the ring of Gaussian integers. ( - 9 i + 8, i + 18 ) = a + b i

A:

Q: 3. Let R be a commutative ring. Let x, y E R such that their characteristics are p. Show that (x +…

A:

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

A:

Q: a. Complete the below chart of ring ( image 2 ), providing the values for a, b, c, d ,e f, g, h, i,…

A: Given: Addition and multiplication table of ℤ8. To find: a. Values of a, b, c, d ,e, f, g, h, i, j…

Q: Consider the ring (Z, 0, 0) where and are defined as follows a b = a+b-1 aob = a + b - ab (a) Is the…

A:

Q: Let R be a ring. Let p and q be elements of R[x] such that deg(p) = 1 and deg(q) = 2. True or false:…

A: Given that R be the Ring and p and q are elements of R[x] Deg(p)=deg(q)

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

A:

Q: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

A:

Q: 3. Consider the set {a+b√2: a, b E Z} with the usual addition and multiplication. Show that this is…

A:

Q: The zero ring ⟨{0}, #, ∗⟩ is a field with respect to addition # and multiplication ∗ because it…

A: The given Statement is  the Zero ring is a field .

Q: Let R = {2n: n E Z} and define addition and multiplication O in R by a b = a + b and aOb = for all…

A:

Q: Define the binary operations + and . On Z by a +b = a+b-1 and a.b= a+b- ab for all ab belongs to Z.…

A: We will use basic properties of ring homomorphism and integral domain.

Q: Let R be a ring with unity. Assume b1a=1R and ab2=1R. Prove that b1=b2.   Also, let R be a ring…

A: Ring with unity,so 1R is different from 0R ,

Q: Let R be a commutative ring and a e R. Show that if a is nilpotent, then ab is nilpotent for each be…

A: Solution. Since a is nilpotent , an = 0 for some n ∈ N.

Q: Let R be a ring with unity and assume p, q, r ∈ R∗ Find (pqr)-1and also prove that it’s the inverse…

A:

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: よn aring of (2,+) find In <97

A:

Q: If R is a noncommutative ring with unity and x, y ∈ R, compute the product x(x + y)(x − y)y.

A:

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

A:

Q: If R is a ring, then prove that (R, A, n) is a commutative ring in the algebraic sense.

A:

Q: Determine if the set {x + y∛2 : x,y ∈ Z} is a ring. If so, please prove why. (Z represents integers)

A:

Q: Let R be a ring and a, b in R. Show that ab is nilpotent implies that ba is nilpotent.

A: We know that An element "a" in a ring R is called nilpotent if an = 0 for some positive integer n.…

Q: Define a new addition ⊕ and multiplication ⊙ on ℚ by               a ⊕ b = a + b + 1 and a ⊙ b = ab…

A:

Q: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0

A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…

Q: Consider Z[√2] = {a + b√2 | a,b € Z}. Determine whether (Z[√2], +, •) is a ring.

A:

Q: Let R be the set of all functions. It is known that (R,+,×) is a ring where + and × are both…

A:

Q: 4. Let a 0 in the ring of integers Z. Find b = Z such that ab but (a) = (b).

A:

Q: Consider Z3[√√2] = {a+b√2: a,b € Z₁}. What is the multiplicative inverse of 1-√2 in Z3[√21?

A:

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

A: R is ring with unity and I is ideal of R.