Non-Commutative Rings

Algebra

Prop $M_n(F)$ the ring of $n$-by-$n$ **matrices is a non-commutative ring.

Algebra

An algebra is a ring $A$ equipped with a vector space structure over a field $F$ such that the multiplication is compatible with the scalar multiplication. That is, the ring multiplication $\cdot :A\times A\to A$ is bilinear: for all $u,v,w\in A$ and $a\in F$ we have $(au+v)\cdot w=a(u\cdot w)+(v\cdot w),w\cdot (au+v)=a(w\cdot u)+(w\cdot v)$ The algebra is called non-associative if the multiplication is non-associative. It is unital if there is an element $1\in A$ such that $1\cdot a=a\cdot 1=a$ for all $a\in A$.

e.g. Matrices with elements in $F$, with scalar multiplication and matrix multiplication are a prominent example of an algebra. Polynomials with scalar and polynomial multiplication are another. One example which will become an important prototype for the structures we study in this course is an algebra constructed from an arbitrary group, called the group algebra:

Group Algebra

Given a group $G$ and a field $F$, the group algebra $FG$ of $G$ over $F$, as a vector space, is given by formal linear combinations of elements of $G$: that is, the vector space with basis $G$. The algebra multiplication given by the bilinear extension of multiplication in $G$.

e.g. The quaternions $\mathbb{H}$ is the subring ($\mathbb{R}$-subalgebra) of $M_2(\mathbb{C})$ generated by the matrices: $\left[\begin{array}{cc}1& \\ & 1\end{array}\right],\left[\begin{array}{cc}i& \\ & -i\end{array}\right],\left[\begin{array}{cc}& 1\\ -1& \end{array}\right],\left[\begin{array}{cc}& i\\ i& \end{array}\right]$

Ideals in Non-Commutative Rings

Def Ideals in Non-Commutative Rings

When the ring is non-commutative we have three variants of an ideal: an abelian subgroup $I\subset R$ **in a ring $R$ **is

• left ideal if $ra\in I$ **for all $a\in I$ **and $r\in R$
• right ideal if $ar\in I$ **for all $a\in I$ **and $r\in R$
• a two-sided ideal if it is both left and right.

e.g. In $M_n(F)$ the left (right) ideals are matrices having some columns (rows) consisting of zeros and the other columns (rows) arbitrary. There are no non-trivial two-sided ideals in $M_n(F)$, it is a simple **ring.