12. Let R² be the set of all pairs of real numbers R² = {(a,b)|a, b = R}. The addition is still coordinate-wise but multiplication is defined by (a1, b₁) * (a2, b2) = (a1ª2 − b1b2, a1b2 + a2b1). The set R² equipped with these operations is a commutative ring. (You don't need to show this.) What is the unity element? Explain. What is the multiplicative inverse for (a, b) ‡ (0,0)? Explain. Find (x, y) such that (x, y)*(x, y) = (-1,0). Explain.

Q: Q6: Define the Boolean ring. Is (Z,+,.) Boolean?

A: .

Q: In the ring Z:[x] 1- The inverse of x² is a) x+1 b) x2 +1 с) х d) x2 2- The value that replaces x is…

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

Q: Q2: (A) Choose the correct answer for each of the following: 1. The cancellation law holds in a. any…

A: Answer 1: Must know:The cancellation laws hold in a ring R if an only if R has no zero divisors

Q: Show that the Vingof CamPlex humbers Cf+so) Can be imbealaling inaring of 2x2 matvices (Mzns

A:

Q: * (10 Let (R, +,) be commutative ring with identity. Then a - b = a-c-b=c ifand only if R has no…

A: True

Q: Define commutative ring.

A: Answer:

Q: Qz: Let Z be the set of integer numbers and *, o defined as follows Va, be Z, then a *b = a +b+1 a…

A: See the attachment

Q: Construct the multiplication table for the ring Z/6Z.

A:

Q: The number of zero divisors of the ring Z4 O Z2 is a) 11 b) 5 c) 7

A:

Q: Enter the multiplicative inverse of the element [21]49 X+ [18]49 in the ring Z49[x].

A: Extended Euclidean Algorithm

Q: Q1) In a ring Z, find the least common multiple of {2,3,8,12).

A: Given problem Given that   In a ring Z , We have to find the Least common multiple of { 2,3,8,12…

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

A:

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: Enter the multiplicative inverse of the element [10] x + [13] in the ring z-[x]

A: We need to fin the multiplicative inverse of the element [10]20x+[13]20 in the ring Z20x. Let…

Q: If a and b are nilpotent element of a commutative ring, show that a+b is also nilpotent. Give an…

A:

Q: Please use method from Abstract Algebra, Show that the polynomial 37a² + 30a³ + a⁴ = 1500 does not…

A:

Q: Explain why the ring of integers Z under usual addition and multiplication is not a field.

A: Given that the set of integers Z is a ring under addition and multiplication. We know that an…

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

A:

Q: Check if the following subsets of real numbers are Rings or not. • {a + b*(2)^(1/3) | a, b ∈ Q} • {a…

A:

Q: The triple ⟨911Z, +, ⋅⟩ forms a commutative ring. It is also an integral domain. However, it is not…

A:

Q: Give an example of a noncommutative ring with unity.

A:

Q: a. Find the elements of Zg that have multiplicative inverses.

A: