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One parking garage offers unlimited parking at a flat rate per month. Another garage offers an hourly rate for parking.
To Emile, because he likes math problems, I leave a choice. He can have one of two inheritances. He must make his choice before leaving the office today.
Let phi be a homomorphism of a ring R with unity onto a nonzero ring R'. Let u be a unit in R. Show that phi (u) is a unit in R'.
Show that every maximal ideal of R is prime and if R is finite, show that every prime ideal is maximal.
If A is an ideal of R and S is a subring of R, then S+A is a subring, A, and (S intersecting A) are ideals of S+A and S.
Show that if U is the collection of all units in a ring with unity, then is a group. A reminder was given to make sure to show that U is closed
Let p be a prime. Show that in the ring Z-p (set of integers modulo p) we have (a+b)^p = a^p+b^p for all a, b in Z-p.
If phi: R1 -> R2 is a ring isomorphism, show that phi extends to an isomorphism phi hat : F1 -> F2.
Find [ 2Z : 12Z ], where 2Z is the ring of even integers. If R is assumed to have a unity, what can you say about [ R : I ] ?
Prove that a module over a polynomial ring C[t] is a finite dimensional vector space with a linear operator that plays the role of multiplication by t.
Now let I be the set of all functions f(x) an element of R such that f(1)=0. Show that I is a maximal ideal of R.
Given two elements a,b in the Euclidean ring R their least common multiple c ? R is an element in R such that a | b and b | c.
Prove that the units in a commutative ring with a unit element form an abelian group.
Prove that a necessary and sufficient condition that the element 'a' in the Euclidean ring is a unit is that d(a) = d(1).
Prove that if M is a maximal ideal of S and phi is surjective then phi^-1(M) is maximal ideal of R.
Show that the field of fractions of the power series ring Z[[x]] is properly contained in the field of Laurent series Q((x)).
Let m and n be positive integers with n dividing m. Prove that the natural surjective ring projection Z/mZ ->Z/nZ is also surjective on the units.
If the least common multiple of a and b in the Euclidean ring R is denoted by [a,b], prove that [a,b] = ab / (a,b) , where (a,b) is the greatest common divisor.
Let p be a prime and let G be an abelian group of order a power of p. Prove that the nilradical of the group ring G is the augmentation ideal.
Let R be a ring such that the only right ideals of R are (0) and R. Prove that either R is a division ring or, that R is a ring.
Prove that I = is not a prime ideal of Z[i]. How many elements are in Z[i] / I. What is the characteristic of Z[i] / I.
Let F be the set of all functions f : R ? R. We know that is an abelian group under the usual function addition, (f + g)(x) = f(x) + g(x).
Let F, K be two fields F ? K and suppose f(x), g(x) ? F[x] are relatively prime in F[x]. Prove that they are relatively prime in K[x].
Let R be a ring with 1 and let S=M2(R). If I is an field of S .show that there is an ideal J of R such that I consists of all 2X2 matrices over J.
Prove that x2 + 1 is irreducible over the field F of integers mod 11 and prove directly that F[x] / (x2 + 1) is a field having 121 elements.