Consider the subset Z[i] = {a + bi | a, b € Z} of the complex numbers, C. Note: Z[i] is called the set of Gaussian integers. a. Prove that Z[i] is a subring of C. b. Is Z[i] a commutative ring? Justify your answer. c. Is Z[i] a ring with identity? Justify your answer. d. Is Z[i] an integral domain? Prove or disprove. [Hint: C is an integral domain.] e. Is Z[i] a field? Prove or disprove. [Hint: The inverse of the complex number a + bi is 1 a-bi = :] a+bi a²+b²

Q: Let Q[√c] denote the set of numbers a + b ⋅√c, where a,b ∈ Q for some c ∈ Z. I must prove that Q[√c]…

A:

Q: Let R be a ring and A be an idea of R where r, sER Explain why rs+ A=sr+A is equivalent to rs – sr E…

A:

Q: Determine if indicated set forms a ring: S = {ri | r E R, i2 = -1} under the operations on complex…

A:

Q: Consider the ring C = {a+ bila, b e R} of complex numbers with respect to standard addition and…

A: Solution:

Q: Show that conjugation respects multiplication, i.e.. C₁ C₂ = C1 C₂.

A:

Q: J. Properties of Divisors of Zero Prove that each of the following is true in a nontrivial ring. 1…

A:

Q: 4. a) What are the units of Z3? b) What are the units of Z4? c) Find the units in Z3 × Z4. d) Look…

A: Step 1: Step 2: Step 3: Step 4:

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: Construct the multiplication table for the ring Z/5Z.

A:

Q: Figure out what the ring Z⁄⟨−4⟩ is. 2- Find a ring that is an integral domain, but not a UFD, that…

A:

Q: The rings Z and 5Z are isomorphic.

A:

Q: et R be a ring and assume a∈R is not a zero divisor. If ab=ac for some b,c∈R, then b=c

A: