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If R is an integral domain with unit element, prove that any unit in R[x] is a unit in R and any unit in R is also a unit in R[x].
Let R be a commutative ring with no non-zero nilpotent elements ( that is, an = 0 implies).
If R is a unique factorization domain and if a, b are in R, then a and b have a least common multiple (l.c.m.) in R.
Prove that if R is an integral domain, then R[x1 , x2, .....,xn] is also an integral domain.
Let R={f(x) included in Q[x]: the coefficient of x in f(x) is 0} Prove that R is a subring of Q[x].
If the sequence is split over S, then it is split over R. If the sequence is split over R, then it is split over S.
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}.