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Rings and Algebras Problem set #3: Solutions Sept. 29, 2011. 1. Find a semiprimitive ring R which has a unique nontrivial twosided ideal.
 

Summary: Rings and Algebras Problem set #3: Solutions Sept. 29, 2011.
1. Find a semiprimitive ring R which has a unique non­trivial two­sided ideal.
Solution. Let R = EndK V , where V is a vector space with dimK V = #0 . We know from Problem 2/2 that
R is semiprimitive (even primitive) and R has a unique non­trivial ideal consisting of endomorphisms of finite
rank.
2. Let V be a (not necessarily finite dimensional) vector space. Find the Jacobson radical of the
exterior algebra #(V ).
Solution. If V is finite dimensional then the ideal I of elements with 0 constant term (note that this ideal is
generated by the image of V in #(V )) is clearly nilpotent. On the other hand #(V )/I # K, hence I = J(#(V )).
If V is infinite dimensional then the ideal generated by V is not nilpotent any more, but it is still a nil ideal.
Thus J(#(V )) is once more the ideal I of elements with 0 constant term.
3. Prove that if R is a principal ideal domain with infinitely many prime elements then J(R) = 0.
Solution. Any such R is a unique factorization domain (UFD). Maximal ideals will correspond to prime
elements, and an element in the intersection of all maximal ideals must be divisible by each of the inifinitely
many prime elements. Hence R is semiprimitive.
4. A submodule N # M is called superfluous in M if for any submodule K # M we have that if
K +N = M then K = M . Prove that J(R) is always a superfluous left ideal in R but give an
example of a module M where rad M is not superfluous. (Recall that the radical of a module
is the intersection of its maximal submodules.)
Solution. Suppose J(R) + K = R for some left ideal K. Suppose K #= R. Since R is finitely generated as

  

Source: Ágoston, István - Institute of Mathematics, Eötvös Loránd University

 

Collections: Mathematics