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A ring with commutative multiplication.
wp-comm-ringstacks-comm-algebrastacks-algebra-textEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Retain the commutative-ring operations and forget the nonzero-invertibility axiom.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
wp-fieldnlab-fieldstacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
A polynomial ring over a commutative coefficient ring is itself commutative.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
milne-fields-galois-theorystacks-algebra-textwp-comm-ringThis is an authored directed relation from the source endpoint to the target endpoint.
Require multiplication to commute.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-comm-ringstacks-algebra-textEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
The functor gives an anti-equivalence between commutative rings and affine schemes; morphisms reverse direction.
Reinterpret an object, pass to an equivalent presentation, or relate canonically corresponding structures; the carrier may change.
stacks-affine-schemewp-affine-schemenlab-affine-schemestacks-schemes-textThis is an authored directed relation from the source endpoint to the target endpoint.
Require descending chains of ideals to stabilize.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
stacks-artinian-ringwp-artinian-ringstacks-schemes-textThis is an authored directed relation from the source endpoint to the target endpoint.
Require and every nonzero element to be invertible; inverse elements are unique.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-fieldnlab-fieldstacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
Ideals of a commutative ring are additive subgroups closed under multiplication by arbitrary ring elements, providing the compatible equivalence data used in quotient rings.
The target represents a state, property, observable, or state-dependent description associated with the source system or theory.
stacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
Require and require a product to vanish only when one factor vanishes.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-integral-domainmilne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Require exactly one maximal ideal.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
stacks-local-ringwp-local-ringnlab-local-ringstacks-schemes-textThis is an authored directed relation from the source endpoint to the target endpoint.
Require every ideal to be finitely generated.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
stacks-noetherian-ringwp-noetherian-ringnlab-noetherian-ringstacks-schemes-textThis is an authored directed relation from the source endpoint to the target endpoint.
Form finite formal sums in a variable with coefficients in the ring.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
milne-fields-galois-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Prime ideals with the Zariski topology and structure sheaf form the affine spectrum.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
stacks-schemes-textThis is an authored directed relation from the source endpoint to the target endpoint.
For an ideal , the quotient identifies exactly when ; the cosets inherit well-defined ring operations.
Identify elements by a stated equivalence relation and equip the quotient with the induced structure. The edge detail must state the representatives and equivalence relation.
stacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
Localization universally makes selected ring elements invertible.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
stacks-algebra-text