covering maps

Let q: X->Y and r:Y->Z be covering maps; let p=(r(q(x))). Show if r^(-1)(z) is finite for each z in Z, p is a covering map.

abelian(fundermental group)

let x0 and x1 be points of the path-connected space X.Show that Pi_1(X,x0) is abelian iff for every pair a and b of paths from x0 to x1, we have a'=b'. where a'([f])=[a-]*[f]*[a];( a- means the reverse of a.) and [f] belongs to Pi_1(X,x0). a':Pi_1(X,x0)->Pi_1(X,x1).

trace on inner-product space

Suppose that V is an inner-product space. Prove that if T: V-->V is a positive operator and trace(T)=0, then T=0.

Boolean Ring

1)Let X={1,2,...,n}and let R be the Boolean ring of all subsets of X. Define f_i:R->Z_2 by f_i(a)=[1] iff i is in a.Show each f_i is a homomorphism and thus f=(f_1,...,f_n):R->Z_2*Z_2*...*Z_2 is a ring homomorphism.Show f is an isomorphism. 2)If T is any ring,an element e of T is called an idempotent provided e^2=e.The elements 0 and 1 are idempotents called the trivial idempotents. Suppose T is a commutative ring and e in T is an idempotent with 0/=e/=1 (/=:is not equal to).Let R=eT and S=(1-e)T.Show each of the ideals R and S is a ring with with identity,and f:T->R*S defined by f(t)=(et,(1-e)t) is a ring isomorphism. 3)Use the result from 2) to show that any finite Boolean ri... click for more

Z-modules

1)I understand what a standard R-module (ring-module)is, but I have heard talk of modules associated with representations. Could someone please give me some idea of what these are? 2) I am trying to find all modules over Z-the Integers; so far, I have only come up with additive groups. How can I find all others?