ifD is an ordered integral domain with positive elements D^p and unity e. prove if a∈D then a>a-e
ifD is an ordered integral domain with positive elements D^p and unity e. prove if a∈D then a>a-e
Suppose (Z, +, ·) is an ordered integral domain. Let a and b and c be elements of Z such that c^2 + a · c + b = 0. (a) Prove that if 4b = a^2 , then x^2 + a · x + b ≥ 0 for all x ∈ Z. (b) Prove that if 4b =/= a^2 , then there is exactly one element d in Z such that d^2 + a · d + b =...
8, Prove that if R is an integral domain and a E R such that a2 + 2a + 1-0 then a =-1. Then give an example of a ring that is not an integral domain for which a2a 1-0 but a f -1. 8, Prove that if R is an integral domain and a E R such that a2 + 2a + 1-0 then a =-1. Then give an example of a ring that is not an integral domain...
8. Let A be an integral domain containing elements x, y, and z. Prove the following facts. (a) If z|x and zly, then x/2 + y/2 = (x + y)/2. (b) If 2 x, then y. (x/2) = (y • x)/2. (c) If yız and x[(z/y), then (x • Y)|z, and 2/(x • y) = (z/y)/x.
A continuous probability density function is a non-negative continuous function f with integral over its entire domain D R" equal to unity. The domain D may have any number n of dimensions. Thus Jpfdzi..d 1. The mean, also called expectation, of a function q is denoted by or E(a) and defined by J.pla f)dy...dr The same function f may also represent a density of matter or a density of electrical charges. Definition 1 The Bivariate Cauchy Probability Density Function f...
7. Can you construct an integral domain with exactly 4 elements? If so, do it. If not, explain why not. Note: when constructing rings, you generally use two Cayley tables; one for each operation. 8. Prove that if R is a Boolean ring with more than two elements, R is not a field.
First: As I mentioned in my e-mail, a Euclidean valuation on an integral domain R is a function u : R* → N (where R* is the set of nonzero elements of R, and N includes 0) with two properties: (1) if a,b E R*, thern (a) v(ab); and (2) if a, b R and b 0, then there exist elements q,r R such that a-bqr and either 0 or v(r) < v(b). Prove that if o is a Euclidean...
37. Show that if D is an integral domain, then 0 is the only nilpotent element in D. 38. Let a be a nilpotent element in a commutative ring R with unity. Show that (a) a = 0 or a is a zero divisor.. (b) ax is nilpotent for all x ER. (c) 1 + a is a unit in R. (d) If u is a unit in R, then u + a is also a unit in R.
1,(Z) = { a bla, b, c, d. Let M2(Z) = a, b, c, d e Z} with matrix addition and multiplication. Which of the following is true: it is a commutative ring with unity it is a ring with unity but not commutative it is a ring without unity and not commutative it is a commutative ring without unity Question 4 Which of the following statement is true about the ring of integers (with usual addition and multiplication) the...
Please help! Thank you so much!!! 2. (10 points) Let R be an integral domain and M a free R-module. Prove that if rm 0 or m 0 where r E R and m E M, then either r 0. 2. (10 points) Let R be an integral domain and M a free R-module. Prove that if rm 0 or m 0 where r E R and m E M, then either r 0.
88. Let D be an integral domain. (a) For a, b E D define a greatest common divisor of a and b. (b) For rE D denote (x)dr dE D.Prove that if (a) +(b)- (d), then d is a greatest common divisor of a and b. 88. Let D be an integral domain. (a) For a, b E D define a greatest common divisor of a and b. (b) For rE D denote (x)dr dE D.Prove that if (a) +(b)-...