special notations in algebra

• $[G:H]$,   index (http://planetmath.org/Coset) of a subgroup $H$ in the group $G$

• $H◁G$,   $H$ is a normal subgroup of $G$

• $G/H$,   quotient group of $G$ with respect to the normal subgroup $H$

• $R/I$ or $R-I$,   quotient ring or difference ring of $R$ with respect to the ideal $I$

• $\mathrm{Z}(G)$,   center of the group $G$

• $R[G]$,   group ring of the group $G$ over the ring $R$

• $R[\alpha ]$,   ring adjunction

• $K(\alpha )$,   field adjunction

• $S^{-1}R$,   localization of the ring $R$ at $S$

• $R_{𝔭}$,   localization of the ring $R$ at the complement of the prime ideal $𝔭$

• $(R,𝔪)$,   local ring $R$ with its maximal ideal $𝔪$

• ${}_{R}M$,   a left $R$-module (http://planetmath.org/Module) $M$

• $M_R$,   a right $R$-module (http://planetmath.org/Module) $M$

• $K/k$,   field extension where $K$ is an extension field of $k$

• $[K:k]$,   degree (http://planetmath.org/ExtensionField) of the field extension $K/k$

• $\overline{K}$,   algebraic closure of a field $K$

• $R{\alpha }_1+\mathrm{\dots }+R{\alpha }_n$  or  $⟨{\alpha }_1,\mathrm{\dots },{\alpha }_n⟩$,   left ideal of the ring $R$ generated by (http://planetmath.org/IdealGeneratedByASet) ${\alpha }_1$, …, ${\alpha }_n$

• $A_1\cong A_2$,   isomorphism (http://planetmath.org/Isomorphism) between the systems (http://planetmath.org/AlgebraicSystem) $A_1$ and $A_2$