Recent content by xixi

Let R be a finite commutative ring . Then show that each element of R is a unit or a zero-divisor .

the subjects were about "zero-divisor graphs of rings "and the graphs associated to rings . Actually I look for a subject in ring theory .

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...

he suggested some subjects to me that aren't interesting to me .

Hi 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 !

Do you mean that RxF is a counter-example ? but RxF has infinite non-unit elements .

so then??...

Let R be a non-commutative ring . Suppose that the number of non-units of R is finite . Can we say that R is a finite ring?

Notice that E is not the identity element because its nth row and nth column are zero .

Let R = M[SIZE="1"]n(F) be the ring consists of all n*n matrices over a field F and E = E[SIZE="1"]11 + E[SIZE="1"]22 + ... + E[SIZE="1"]n-1,n-1, where E[SIZE="1"]ii is the elementary matrix (E[SIZE="1"]ij is matrix whose ij th element is 1 and the others are 0) . I know that RE is a maximal...

Let R = M[SIZE="1"]n(F) the ring consists of all n*n matrices over a field F and E = E[SIZE="1"]11 + E[SIZE="1"]22 + ... + E[SIZE="1"]n-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...

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...

of course 0 is a zero-divisor and belongs to D(R).

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...