Let Rx denote the group of nonzero real numbers under multiplication and let R+ denote the...

Let R^x denote the group of nonzero real numbers under multiplication and let R⁺ denote the group of positive real numbers under multiplication. Let H be the subgroup {1, −1} of R^x. Prove that R^x ≈ R⁺ ⊕ H.

Expert Solution

Colby Messinger answered 2 years ago

Suppose we define a relation ~ on the set of nonzero real numbers R* = R\{0}...

Suppose we define a relation ~ on the set of nonzero real numbers R* = R\{0} by for all a , b E R*, a ~ b if and only if ab>0. Prove that ~ is an equivalence relation. Find the equivalence class [8]. How many distinct equivalence classes are there?

Due October 25. Let R denote the set of complex numbers of the form a +...

Due October 25. Let R denote the set of complex numbers of the form a + b √ 3i, with a, b ∈ Z. Define N : R → Z≥0, by N(a + b √ 3i) = a 2 + 3b 2 . Prove: (i) R is closed under addition and multiplication. Conclude R is a ring and also an integral domain. (ii) Prove N(xy) = N(x)N(y), for all x, y ∈ R. (ii) Prove that 1, −1 are the...

Let D10 denote the dihedral group of the hexagon. Thus, D10 is generated by r and...

Let D10 denote the dihedral group of the hexagon. Thus, D10 is generated by r and f with r10=f2=1 and fr=r-1f=r9f (a) Show that D10 has a subgroups N and M such that i. N ∼= D5 (isomorphic to D5) ii. M is a cyclic subgroup group of order 2 iii. N ∩ M = {e} iv. N and M are each normal in D10 v. Every element in g ∈ D10 is a product g = nm of elements...

Let S be the set of all ordered pairs of real numbers. Deﬁne scalar multiplication and...

Let S be the set of all ordered pairs of real numbers. Deﬁne scalar multiplication and addition on S by: α(x1,x2)=(αx1,αx2) (x1,x2)⊕(y1,y2)=(x1 +y1,0) We use the symbol⊕to denote the addition operation for this system in order to avoid confusion with the usual addition x+y of row vectors. Show that S, together with the ordinary scalar multiplication and the addition operation⊕, is not a vector space. Test ALL of the eight axioms and report which axioms fail to hold.

Let R and S be rings. Denote the operations in R as +R and ·R and...

Let R and S be rings. Denote the operations in R as +R and ·R and the operations in S as +S and ·S (i) Prove that the cartesian product R × S is a ring, under componentwise addition and multiplication. (ii) Prove that R × S is a ring with identity if and only if R and S are both rings with identity. (iii) Prove that R × S is a commutative ring if and only if R and...

Let k ≥ 2. Use that R (the real numbers) is complete to show R^k is...

Let k ≥ 2. Use that R (the real numbers) is complete to show R^k is complete.

Let Z* denote the ring of integers with new addition and multiplication operations defined by a...

Let Z* denote the ring of integers with new addition and multiplication operations defined by a (+) b = a + b - 1 and a (*) b = a + b - ab. Prove Z (the integers) are isomorphic to Z*. Can someone please explain this to me? I get that f(1) = 0, f(2) = -1 but then f(-1) = -f(1) = 0 and f(2) = -f(2) = 1 but this does not make sense in order to...

If p is prime, then Z*p is a group under multiplication.

A sequence is just an infinite list of numbers (say real numbers, we often denote these...

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Let R be a ring (not necessarily commutative), and let X denote the set of two-sided...

Let R be a ring (not necessarily commutative), and let X denote the set of two-sided ideals of R. (i) Show that X is a poset with respect to to set-theoretic inclusion, ⊂. (ii) Show that with respect to the operations I ∩ J and I + J (candidates for meet and join; remember that I+J consists of the set of sums, {i + j} where i ∈ I and j ∈ J) respectively, X is a lattice. (iii) Give...

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