1. Prove: The square of any integer n is of the form 3k or of the form 3k+1 for some integer k. 2. Let a be an element of a ring R such that a3=1R. Prove: for any integer n, either (an)n=1R or (an)n=a.

Q: Consider the following closed semi-ring: (R₂ U{+∞o), Min, Max, +∞0, 0) where Min is the operation +…

A: It is given that the closed ring , where Min is the operation + and Max is  an operation.The…

Q: Let n > 2 be an integer. Show that Z/Zn is a field if and only if n is a prime number.

A: Let A=1¯,2¯,3¯,...,n-1¯ we show for ⇒ Let ℤℤn is a field⇒∀a∈A,∃x∈A such that ax=1¯ax≡1modn has a…

Q: Q2. Recall the ring of infinitesimals C[e] that was introduced in the first lecture. Find all units…

A: Cε=Rε∈Cε |  R ε is polynomial in ε Let R be any Ring. 0≠x∈R is said to be unit if there exist…

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

A:

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: 2. Find all values of p such that Z„[r]/{x² + 1) is a field.

A: We know that results

Q: Define the concept of field. Is Za field?

A: Definition of Field:-A commutative ring (R ,+,.) with unity is called field if each non zero element…

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

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: (b) Let I, J be ideals (two-sided) of a ring R and check as to whether In J is an ideal (two-sided)…

A:

Q: Let be the following set of rational numbers: Consider the algebraic structure (R2, +, ·), where +…

A: Field :-  A commutative ring F with unity is called field, if each nonzero elements of F has…

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: Let a be an element of a ring R such that a3=1R. Prove: for any integer n, either (an)n=1R or…

A: Let a be an element of a ring R such that a3=1R.  We will find, for any integer n, either (an)n is,…

Q: 1. Find all units in Z[V-3]. Justify your answer. 2. Prove that 2 is an irreducible element in…

A:

Q: Prove that in a Euclidean ring R, (a, b) can be found as follows : b = q0 a + r¡, where d (r¡)<d (a)…

A:

Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

A:

Q: For any prime number p and any positive integer n, prove that every zero in the ring Z-n is…

A: We need to show that the every zero element in the ring is nilpotent element.

Q: For an arbitrary field K and an odd positive integer n, show that K(2n) K(n)

A: To show thatK K2n=Kn we need to prove that the set of all polynomials of degree 2n over is K equal…

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: Q29: Define the Boolean ring. Is (Z,+,.) Boolean?

A:

Q: Let R = ({0, 1,,8), 0, 0) be a ring where and are defined as follows: a b = (a + b) mod 9 a Ob= (ab)…

A: Using the definition of the Ring R we answer all the questions. Important thing to note is the using…

Q: 2. We say that an element z E F is a square root of a E F whenever 2² = a, we denote the square root…

A:

Q: Which one of the following is true? In the ring Zp, if x = x then x = 0 or x = 1 In the ring Z6, 4…

A:

Q: In a ring R, if there is a solution to the equation ax = 1, then a is called a(n) O primitive…

A: Primitive Element: A primitive element of a finite field is a generator of the multiplicative group…

Q: Q2: Let Z be the set of integer numbers and , defined as follows Va, be Z, then ab = a + b+1 aob =…

A: