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Let $\mathbb{C}_2$ denote the group of order $2$ and let $\mathbb{Z}_4$ denote the ring of integers modulo $4$. The group ring $R = \mathbb{Z}_4\mathbb{C}_2$ is commutative.

My problem is how to generate all the elements of this ring and the operations involved. I am a learner try to teach myself something to do with group rings.

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Let's denote by $0,1,2,3$ the elements of $\mathbb{Z}_4$ and by $\mathbf{1}$ (the identity) and $\mathbf{u}$ the elements of $C_2$. An element of the group ring is of the form $$ a\mathbf{1}+b\mathbf{u} $$ where $a,b\in\mathbb{Z}_4$. So we have sixteen elements in all. The zero element is $0\mathbf{1}+0\mathbf{u}$, the identity is $1\mathbf{1}+0\mathbf{u}$.

Addition is performed in the obvious way: $$ (a\mathbf{1}+b\mathbf{u})+(c\mathbf{1}+d\mathbf{u})= (a+c)\mathbf{1}+(b+d)\mathbf{u} $$ Multiplication is performed with the usual rules: $$ (a\mathbf{1}+b\mathbf{u})(c\mathbf{1}+d\mathbf{u})= ac\mathbf{1}\mathbf{1}+ad\mathbf{1}\mathbf{u}+b\mathbf{u}\mathbf{1}+ bc\mathbf{u}\mathbf{u}= (ac+bd)\mathbf{1}+(ad+bc)\mathbf{u} $$ because $\mathbf{u}\mathbf{u}=\mathbf{1}$ in $C_2$.

Note that we haven't used any special property of $\mathbb{Z}_4$, just that it is a ring. By definition, the elements of the group (in this case $C_2$) commute with the elements of the ring (here $\mathbb{Z}_4$).

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