The set of equivalence classes Zn = {[0], [1], - - · , [n – 1]} is an example of what is...called a finite ring (which simply means the elements can be added and multiplied in familiarways).(Def.) Two non-zero elements a and b in a ring are called zero divisors if a·b = 0. For example,in Z12 the elements [2] and [6] are zero divisors. Prove the following theorem.Theorem. If n is composite, then there exists at least one pair of zero divisors in Zn.

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Answered: The set of equivalence classes Zn =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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2. Define a set X as X = RU {-}. Define two binary operations over X as follows: Addition:x O y = max (x, y) Multiplication: xy = x + y (here "+" denotes the usual addition or real numbers) investigate the properties of a commutative ring for this structure; that is, write proofs for the properties that hold or give an explanation on why a property does not hold if it doesn't What kind of algebraic structure is this? Note: The properties of -oare the same as what you would have seen in Calculus courses.
4. Write out the addition and multiplication tables for each congruence class ring below, then determine if it is a field. (a) Z3[r]/(x² +1) (b) Z₂[r]/(x² +1)
Let R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a zero-divisor. We start the proof by (ab) e = 0, e# 0. Which of the following is a true statement in the proof? If ac = 0 then a = 0 If ac + 0 then b = 0 If ac = 0 then a is a zero divisor If bc = 0 then a = 0
In the set C of ordered pairs of real numbers, addition and multiplication are defined by the formulae (a, b)+(c, d)=(a + c, b +d) and (a, b).(c, d)=(ac - bd , ad + bc). Show that the system (C,+, .) is a field. What are the additive and multiplicative identities of of this field ?
1) Let R and S be rings. Show that the cartesian product of sets R x S = {(r, ) |r e R, s €s} with the addition (r, s) + (r', s') := (r +r', s + s') and the multiplication (r, s) · (r', s') := (r - r', s - s') is a ring.
2. Let R be a ring, and a, b e R. Prove exactly one of the following statements: (a) If ab is a left zero divisor, then either a or b must be a left zero divisor. (b) If a is nonzero but not a left zero divisor, then a may be cancelled on the left.
Enter the multiplicative inverse of the element [22]44 X+ [25]44 in the ring Z44[x]. When you enter your answer, make sure that all congruence classes are in the form [a]44 with 0 ≤ a<44. You don't have to type the brackets. For example, if your answer is [2]44 x² + [3]44, you can type 2x^2+3. Answer:
Demonstrate or explain why the system (P(R#), +, *) is NOT a field, that is, demonstrate or explain why some elements in set P(R#) do not have inverses corresponding to the * operation.
3. Let A and B be rings such that A and B are reducible modules. Let R = A B be the ring direct sum of A and B. Show that RR is reducible.
Give an example where a and b are not zero divisors in a ring R, but the sum a + b is a zero divisor.
Find a bijection from the set of positive integers Z+ to the set of integers Z (Thus, conclude Z+ = Z)
Let S = {4, 7, 41, 74}. Recall that the power set P (S) of S is the set of all subsets of the set S. Notice that (P (S);A,n) is a commutative ring with unity, where AAB = (A – B)U (B – A) for any A, BE P (S) and -, U,n are the usual set difference, union, and intersection, respectively. List the elements of the following sets: a. The additive identity, which is the element with respect to A, is the set b. The the additive inverse of A = {4}, which is the inverse of A with respect to A, is the set c. The unity is the set d. The set S = {4,7, 41, 74} is ? v since the additive identity the unity e. The set A = {4} is ? v since An ? f. The ring (P (S);A,n) is ?

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