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Show that the class of all finite unions of sets of the form A X B with A ? A and B ? B is a ring of subsets of X ? Y .
Prove that [O:Of]= f (index as additive abelian groups). Prove conversely that a subring of O containing the identity.
Prove conversely that if R is a commutative ring with 1 in which the set of non-units forms an ideal M, then R is a local ring with unique maximal ideal M.
Let R = C([0, 1]) be the ring of continuous real-valued functions on the interval [0, 1], with the usual definitions of sum and product of functions.
Let R be a domain. Prove that the polynomial f(x) is a unit in R[x] if and only if f(x) is a nonzero constant which is a unit in R.
More generally, show that if a divides bc with nonzero a,b then a/(a,b) divides c. (Here (a,b) denotes the g.c.d. of a and b).
A local ring is commutative with identity which has a unique maximal ideal. Prove that R is local if and only if the non-units of R form an ideal.
Give a direct proof (i.e. without appealing to part c) that if R is an integral domain, then it is a field.
Show that the inner radius of ring A is squared root 60cm, and find similar expressions for the inner radii of ring B and ring C.
Find elements x and y in a ring of characteristic 4 such that (x + y)^4 != x^4 + y^4.
Indicate which of the following statements about sets under the specified operations are true. For the ones that are false, provide one counter-example.
How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP ring is $30 and on an SST ring is $60?
Prove that a finite subring R of a field F is itself a field. Hint: if x is an element of R and x is not equal to 0 show the function f:R->R with f(r).
If R and S are rings, the cartesian product RxS is a ring too with operations and additive inverse -(r,s) = (-r,-s)
Suppose F, E are fields and F is a subring of E. Suppose g is an element of E and is algebraic over F, that is p(g)=0 for some nonzero.
If E is a field and H is a subgroup of Aut(E), show that the set E^H of elements of E that are fixed (sent to themselves) by every Q in H.
If a,b are positive integers with greatest common divisor d and least common multiple m, show aZ+bZ=dZ and aZ intersect bZ=mZ.
Let R be a commutative ring and let I ? R be an ideal. Show that vI := { f ? R | there exists n ? N such that fn ? I } is an ideal of R.
We have seen that R is a principle ideal domain. That is, every ideal is generated by a single element of R. Find h(x) in R so that I = {f(x)h(x) | f(x) in R}.
Let p be any prime integer. Consider polynomials f(x) and g(x) of the form. Consider the multiplicative group of nonzero elements of Zp.
How to evaluate personality theory over time, and how and why historical personality theories lend themselves to understanding people and events.
If I,J are ideals in a ring R such that I+J=R and R is isomorphic to the product ring (R/I)x(R/J) when IJ=0, describe the idempotents corresponding.
In each of the cases below, describe the ring obtained from F2 by adjoining an element x satisfying the relation: (i) x2+x+1=0, (ii) x2+1=0, (iii) x2+x=0.
For the previous rings that you decided are not local, slightly change their definition (without proof) such that you get a local ring.
Identify Hom(Z/nZ,Z), Hom(Z,Z/nZ), Hom(Z/3Z,Z/6Z), Hom(Z/10Z,Z/6Z) as abelian groups, where n belongs to Z and Z is the set of integers.