Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: nonunital ring

Bottom of Page

1 to 4 of 4

  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 4th 2023

    Added:

    A survey of various notions between unital rings and nonunital rings:

    • Patrik Nystedt, A survey of s-unital and locally unital rings, Revista Integración, temas de matemáticas Escuela de Matemáticas, Universidad Industrial de Santander, Vol. 37, N◦ 2, 2019, pág. 251–260. doi.

    diff, v21, current

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeSep 4th 2023
    • (edited Sep 4th 2023)

    I do not quite understand why unitisation is treated for commutative rings here. In my recollections, formally adding a unit is useful when treating module categories over general (=nonunital) rings as categories of unital modules over associated unital rings. (Of course, the category of nonunital modules over the associated unital ring are different.)

    However there are some important differences between unital and nonunital rings. First of all, in the unital case every Artinian ring is Noetherian, but not other way around. This is a bit misterious at a first glance, but the true reason of breaking the symmetry is the existence of the unit element: the fact does not extend to the nonunital context.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 5th 2023

    I don’t see that removing the word “commutative” throughout would break any of the propositions (esp. Propositions 3.6 and 3.8), so I think it would be fine to remove it. My guess is that whoever wrote it in was simply more familiar with the commutative case.

  4. changed higher algebra - contents to algebra - contents in context sidebar

    Anonymouse

    diff, v23, current

1 to 4 of 4

  1.