Recent content by rachellcb

Thanks! I think I can,... does it involve using the fact that a gcd of f and g is the smallest degree polynomial that can be written as a linear combination of f and g, and that d, being a common factor of f and g in L[x], must divide h?

Thanks, I see that φ(m) = φ(m1) + φ(m2) ⇔ φ(m1) = φ(m) - φ(m2) ⇔ φ(m1) = φ(m - m2) ⇔ φ(m1) = φ(m - φ(m)) But neither m1 nor m need be in the image of φ, so injectivity does not necessarily apply, right?

Homework Statement Let K \subseteq L be fields. Let f, g \in K[x] and h a gcd of f and g in L[x]. To show: if h is monic then h \in K[x]. The Attempt at a Solution Assume h is monic. Know that: h = xf + yg for some x, y \in K[x]. So the ideal generated by h, (h) in L[x] equals...

Homework Statement Let R be a unital commutative ring. Let M be an R-module and \varphi : M \rightarrow M a homomorphism. To show: if \varphi \circ \varphi = \varphi then M=ker(\varphi)\oplus im(\varphi) The Attempt at a Solution I have already shown that M=ker(\varphi)\cap im(\varphi)...

Hey I have also been working on this problem- got as far as showing that if ua and ub \in R such that ua(1-ra) =1=(1-ra)ua and ub(1-rb) =1=(1-rb)u then ubua(1-r(a+b))=1. But (1-r(a+b))ubua=(1-raub)a\neq?1 Does anyone have suggestions on how to go from here?

Homework Statement Let R be a ring with multiplicative identity. Let a, b \in R. To show: 1-ab is a unit iff 1-ba is a unit. The Attempt at a Solution Assume 1-ab is a unit. Then \exists u\in R a unit such that (1-ab)u=u(1-ab)=1 \Leftrightarrow u-abu=u-uab \Leftrightarrow abu=uab. Not...

Solved, thanks! :smile:

Homework Statement Let R be a ring with multiplicative identity. Let u \in R be a unit and let a1, ..., ak be nilpotent elements that commute with each other and with u. To show: u + a1 + ... + ak is a unit. The Attempt at a Solution Need to show that u'(u + a1 + ... + ak)=1 for some u'...

@ 3.141... I can definitely relate to the feeling you described. Often I find it helps to collect all the definitions, theorems, lemmas etc on a particular object given in the problem in one place, and then draw on them to solve the problem. After a while they start to form part of your...

If R is a finite ring of of order p where p is prime, show that either R is isomorphic to Z/pZ or that xy=0 for all x,y in R I know that both R and Z/pZ have the same number of elements (up to equivalence) and that R isomorphic to Z/pZ implies R must be cyclic (I think) but am otherwise...