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rschwieb

PhD in Ring and module theory

If you also enjoy ring theory, you might enjoy the Database of Ring Theory.

Other interests: Clifford algebra/Geometric algebra, Applications of abstract algebra, geometry, Python programming and algorithm design.

Notes for myself:

Let $m$ and $n$ be integers in the ring of integers. Show that $m\mathbb Z$ contains $n\mathbb Z$ if and only if $m$ divides $n$

Prove $R\times S$ is not an integral domain

Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field

Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.

Is the ideal $I = \{f\mid f (0) = 0\}$ in the ring $C [0, 1]$ of all continuous real valued functions on $[0, 1]$ a maximal ideal?

If a ring element is right-invertible, but not left-invertible, then it has infinitely many right-inverses.

Prove that $A+I$ is invertible if $A$ is nilpotent

All group homomorphism from $ \mathbb{Z} _m $ to $\mathbb{Z}_n $

Right and left inverse

Order of general- and special linear groups over finite fields.

If $AB = I$ then $BA = I$

A linear operator commuting with all such operators is a scalar multiple of the identity.

In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.

What are the left and right ideals of matrix ring? How about the two sided ideals?

A ring element with a left inverse but no right inverse?

Explaining the product of two ideals

What's the motivation of the definition of primary ideals?

A ring $R$ which can be written as a direct sum of two ideals.

Ideal contained in a finite union of prime ideals

Proving $a+a =0$ for Boolean ring

$R^n \cong R^m$ iff $n=m$

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