PRIME RINGS SATISFYING A POLYNOMIAL IDENTITY is still direct
... own lower nil radical. We remark that a prime ideal, in fact any ideal modulo which there are no nilpotent ideals, is an algebra ideal, so that a prime quotient R of A is also a polynomial identity algebra in which every element is a sum of nilpotents. But by the last part of the theorem, (Pi = Ç), ...
... own lower nil radical. We remark that a prime ideal, in fact any ideal modulo which there are no nilpotent ideals, is an algebra ideal, so that a prime quotient R of A is also a polynomial identity algebra in which every element is a sum of nilpotents. But by the last part of the theorem, (Pi = Ç), ...
Solutions - Dartmouth Math Home
... elements in Q, we just need to figure out when the multiplicative inverse is contained in R. If a/b ∈ Q is nonzero, then (a/b)−1 = b/a. Therefore R× = {a/b ∈ R : b/a ∈ R}. Writing a/b in reduced form, we must have that both a and b are odd. Therefore R× is the multiplicative group of fractions whose ...
... elements in Q, we just need to figure out when the multiplicative inverse is contained in R. If a/b ∈ Q is nonzero, then (a/b)−1 = b/a. Therefore R× = {a/b ∈ R : b/a ∈ R}. Writing a/b in reduced form, we must have that both a and b are odd. Therefore R× is the multiplicative group of fractions whose ...
2008-09
... Let I, J be ideals of a ring R such that I + J = R. Prove that I J = IJ and if IJ = 0, then R ; R R . ...
... Let I, J be ideals of a ring R such that I + J = R. Prove that I J = IJ and if IJ = 0, then R ; R R . ...
LOCAL CLASS GROUPS All rings considered here are commutative
... and we may form Rp = S −1 R = { as : a ∈ A, s ∈ S} addition and multiplication making sense because we can find common denominators. This process is called localization at p because pRp is the unique maximal ideal in Rp . Example 1.2. (a) Any field k is a local ring, since the only proper ideal (0) ...
... and we may form Rp = S −1 R = { as : a ∈ A, s ∈ S} addition and multiplication making sense because we can find common denominators. This process is called localization at p because pRp is the unique maximal ideal in Rp . Example 1.2. (a) Any field k is a local ring, since the only proper ideal (0) ...
2.1 Modules and Module Homomorphisms
... A-modules correspond to ring homomorphisms from A into endomorphism rings of abelian groups. Examples: (1) If I A then I becomes an A-module by regarding the ring multiplication, of elements of A with elements of I , as scalar multiplication. ...
... A-modules correspond to ring homomorphisms from A into endomorphism rings of abelian groups. Examples: (1) If I A then I becomes an A-module by regarding the ring multiplication, of elements of A with elements of I , as scalar multiplication. ...
