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Abstract Algebra
Let J be an ideal in any ring R. Suppose J^n+1 = 0 (so in particular, J ⊆ rad(R)). Show that 1 + J is a nilpotent group of class ≤ n; that is, 1 + J has a central series of length ≤ n.
Abstract Algebra
Let J be an ideal in any ring R. If J ⊆ rad(R), show that, for any i ≥ 1, the multiplicative group (1 + J^i)/(1 + J^(i+1)) is isomorphic to the additive group J^i/J^i+1.
Abstract Algebra
Construct a commutative noetherian rad-nil ring that is not Hilbert.
Abstract Algebra
Show that: any commutative ring is a quotient of a commutative rad-nil ring.
Abstract Algebra
Let us call a ring R “rad-nil” if its Jacobson radical is a nil ideal. Show that:any commutative artinian ring is Hilbert;
Abstract Algebra
Let us call a ring R “rad-nil” if its Jacobson radical is a nil ideal. Show that:a commutative ring is Hilbert iff all of its quotients are rad-nil.
Abstract Algebra
Let A = R[T], where T is an infinite set of commuting indeterminates. Show that rad A is a nil ideal.
Abstract Algebra
Let R be a graded ring. Show that J = rad R is a graded ideal of R, in the sense that J has a decomposition
J = J0 ⊕ J1 ⊕• • • , where Ji = J ∩ Ri.
J = J0 ⊕ J1 ⊕• • • , where Ji = J ∩ Ri.
Abstract Algebra
A CERTAIN SUM OF MONEY AT COMPOUND INTEREST BECOMES Rs.7,396 IN TWO YEARS AND Rs.7,950.70 IN 3 YEARS.FIND THE RATE OF INTEREST.
Abstract Algebra
James bought a few hamsters. For each day after the first day of the week,the hamsters ate 20 grams of food more than the previous day. The hamsters grew fast, finishing, 1,260 grams of food in the first week. How much food did the hamsters eat on the first day?
