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Abstract Algebra
Show that the following relation C defined on R is an order relation by checking the three axioms.
x C y if |x| > |y| or if |x| = |y| and x > y.
x C y if |x| > |y| or if |x| = |y| and x > y.
Abstract Algebra
Let ф: R ----> R* , where R is additive and R* is multiplicative, by ф(x) = 2^x . prove that ф is an isomorphism or not.
Abstract Algebra
Let фi : Gi ------> G1 x G2 x G3 x…….Gi x ……Gr be given by фi (gi)= (e1 ,e2 ,…..gi,…..er) where gi єGi and ej is the identity of Gj. Prove that this is a injective map.
Abstract Algebra
Let F be an additive group of all continuous functions mapping IR into IR. Let IR be the additive group of real numbers, and let ф :F ------> IR be given by ф(f) = ∫_0^4▒〖f(x)dx〗 . Prove that f is a homomorphism?
Abstract Algebra
Describe the kernel of homomorphism ф: S3 ---->Z2
Abstract Algebra
Show that ф : C ----> M2 (IR) given by
Ф (a+ib) = (■(a&b@-b&a))
For a, b є IR gives an isomorphism of C with the subring Ф[C] of M2 (IR).
Ф (a+ib) = (■(a&b@-b&a))
For a, b є IR gives an isomorphism of C with the subring Ф[C] of M2 (IR).
Abstract Algebra
Find the subring of the ring Zx Zthat is not an ideal of Zx Z
Abstract Algebra
Let R be a commutative domain that is not a field. Show that not always R is not J-semisimple implies R is semilocal, if R is a noetherian domain.
Abstract Algebra
Let R be a commutative domain that is not a field. Show that if R is not J-semisimple then R is semilocal, if R is a 1-dimensional noetherian domain.
Abstract Algebra
Let R be a commutative domain that is not a field. If R is semilocal, show that R is not J-semisimple.
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