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A function between two rings that respects both the addition and multiplication operations. For

A vector space equipped with a bilinear product. It is a set A along with two operations (usually addition and multiplication), satisfying certain axioms.

A set G with a binary operation * satisfying closure, associativity, identity, and invertibility.

A set F with two operations + and *, for which (F, +) is an Abelian group, (F \ {0}, *) is an Abelian group, and * distributes over +.

A set V over a field F with two operations: vector addition and scalar multiplication, which satisfy eight axioms such as associativity, commutativity, and distributivity.

A function between two fields that is both a group homomorphism for addition and multiplication, respecting the operations and additionally the multiplicative identity.

A generalization of vector spaces where the vectors are elements of the module and scalars come from a ring. The module must satisfy conditions analogous to those of a vector space.

A set R with two binary operations + (addition) and * (multiplication) where (R, +) is an Abelian group and * is associative, with distributivity of multiplication over addition.

A function between two groups that respects the group operations. If

A subset I of a ring R such that for any elements $a, b \in I$ and $r \in R$, $(a + b) \in I$ and $(r * a) \in I$.

A subgroup N of a group G such that for every element $g \in G$, the relation $gNg^{-1} = N$ holds, where $gNg^{-1}$ denotes the conjugate of N by g.

An ideal I within ring R such that there is no other ideal J with I strictly contained in J, except for R itself.

A nontrivial group that does not have any proper nontrivial normal subgroups.

An ideal P in a commutative ring R such that if the product $a * b \in P$, then either $a \in P$ or $b \in P$ for $a, b \in R$.

A bijective group homomorphism. It implies both groups have the same structure and that the groups are essentially the same, mathematically speaking.

Given a group G and a normal subgroup N, the quotient group G/N is the set of left cosets of N in G with the group operation defined by the multiplication of cosets.

A group that has a series of subgroups down to the trivial subgroup where each subgroup is normal in the previous subgroup and the quotient groups are Abelian.

A subset S of a field F that is itself a field with respect to the same operations of addition and multiplication defined on F.

An ideal I in a ring R that is generated by a single element a in R, so all elements of I are of the form $r*a$ for some $r \in R$.