G-ring or Grothendieck ring, a type of commutative ring in algebra
It is equipped to handle certain non-commutative algebras which are extensively used in theoretical high energy physics: Clifford algebras, SU(3) Lie algebras, and Lorentz tensors.
Going up and going down, terms in commutative algebra which refer to certain properties of chains of prime ideals in integral extensions
Masayoshi Nagata proved in the 1950s that for any commutative local ring A with maximal ideal m there always exists a smallest ring Ah containing A such that Ah is Henselian with respect to mAh.
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Completeness of a ring is not a necessary condition for the ring to have the Henselian property: Goro Azumaya in 1950 defined a commutative local ring satisfying the Henselian property for the maximal ideal m to be a Henselian ring.
Hilbert C*-modules were first introduced in the work of Irving Kaplansky in 1953, which developed the theory for commutative, unital algebras (though Kaplansky observed that the assumption of a unit element was not "vital").
Hilbert's basis theorem, in commutative algebra, stating every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated
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Hilbert's syzygy theorem, a result of commutative algebra in connection with the syzygy problem of invariant theory
When discretization is commutative with dualization, then, under appropriate conditions, Pontryagin's minimum principle emerges as a consequence of the convergence of the discretization.
Joachim Cuntz is known for his contributions regarding the theory of operator and in the field of the non-commutative geometry, with particular contributions to the structure of simple C*-algebras, the K-theory and cyclic homology.
(This complication is one of the most difficult aspects of Listing's law to understand, but it follows directly from the non-commutative laws of physical rotation, which specify that one rotation followed by a second rotation does not yield the same result as these same rotations performed in the inverse order.)
Regular sequence, which is an important topic in commutative algebra.
Non-abelian group, in mathematics, a group that is not abelian (commutative)
Quaternion, a non-commutative extension of the complex numbers
Vladimir Berkovich reformulated much of the theory of rigid analytic spaces in the late 1980s, using a generalization of the notion of Gelfand spectrum for commutative unital C*-algebras.
Together with Coates, Fukaya, Kato, and Venjakob she formulated a non-commutative version of the main conjecture of Iwasawa theory, on which much foundation of this important subject is based.