Commutative algebra

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• ConceptCommutative ring

  A commutative ring is a ring in which the multiplication is commutative. Commutative algebra is the study of these rings. The definition of a commutative ring gives the precise condition.

• ConceptCommutative algebra over a ring

  A commutative algebra over a commutative ring $R$ is a commutative ring $A$ together with a ring homomorphism $R \to A$. The homomorphism lets one multiply elements of $A$ by elements of $R$, so $A$ is at the same time a ring and an $R$-module. Most rings in commutative algebra and algebraic geometry come with such a structure, for example polynomial rings over $R$ and their quotients. The notion lets one compare rings over a fixed base and speak of $R$-algebra homomorphisms.

• ConceptSubalgebra

  Let $A$ be an associative algebra over a commutative ring $K$. A subalgebra of $A$ is a subset that contains $1$ and is itself a $K$-algebra under the same operations. It is a subring of $A$ that is also a submodule for the scalar multiplication by $K$. Subalgebras are the natural subobjects of algebras. Images of algebra homomorphisms are subalgebras, and so are the subalgebras generated by a few elements, such as $K[a] \subseteq A$. It is the substructure for the operations of a $K$-algebra, with $1$ as a constant.

• ConceptRing homomorphism

  A ring homomorphism is a map between two rings that preserves addition, multiplication and the identity element. The definition of a ring homomorphism gives the conditions.

• ConceptIdeal

  An ideal of a ring is an additive subgroup that is closed under multiplication by every element of the ring. Ideals are the kernels of ring homomorphisms, and they give quotient rings. The definition of an ideal gives the conditions.

• ConceptQuotient ring

  The quotient ring of a ring by a two-sided ideal is the set of the cosets of the ideal. Its operations come from the operations of the ring. The integers modulo $n$ are the main example. The definition of the quotient ring gives the construction.