Let R be a ring . Suppose that e and f=1-e are two idempotent elements of R and we have R=eRe \oplus fRf (direct sum ) and R doesn't have any non-trivial nilpotent element . Set R_1=eRe and R_2=fRf . If R_1=\{0,e\} and R_2 is a local ring , then prove that R_2 is a division ring . (note that e...
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I'm a graduate student in mathematics and abstract algebra, and I'm looking for an exciting subject in algebra to work on for my thesis . I want the subject be new and sth that one can give paper on it . Any suggestion will be appreciated !
Let R = Mn(F) be the ring consists of all n*n matrices over a field F and
E = E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix (Eij is matrix whose ij th element is 1 and the others are 0) .
I know that RE is a maximal left ideal . Let Q be an invertible matrix . Can we say that...
Let R = Mn(F) the ring consists of all n*n matrices over a field F and E = E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix( Eij is a matrix whose ij th element is 1 and the others are 0). Then the
following hold:
1.Every matrix of rank n-1 in any maximal left ideal generates the...
To qualify as a ring the set, together with its two operations, must satisfy certain conditions—namely, the set must be 1. an abelian group under addition; and 2. a monoid (a group without the invertibility property is a monoid) under multiplication; 3. such that multiplication distributes over...
let R be a non-commutative ring and D(R) denotes the set of zero-divisors of the ring . Suppose that z^{2} =0 for any z \in D(R) . prove that D(R) is an ideal of R.
let R be a matrix ring over a finite field \LARGE F_{q} , i.e. \Large R=M_{n}(\LARGE F_{q}). then
1.Every matrix of rank n-1 in any maximal left ideal generates the maximal left ideal.
2.moreover,the number of matrices in every maximal left ideal that can be a generator is the same as the...