A First Course in Noncommutative Rings

Lam, T. Y. (2001). A First Course in Noncommutative Rings. Springer-Verlag.

Type

Book

Intellectual base

Textbook

Year

2001

DOI

10.1007/978-1-4419-8616-0

ISBN

9780387953250

DOI registrant

Springer Nature

Link

https://doi.org/10.1007/978-1-4419-8616-0

Cited by

9 citations across 9 artifacts.

Each citation links to the sentence it supports in the citing article.

Domain-specific

• Domain (ring theory)
  • A domain requires ab equal zero to imply a equal zero or b equal zero; in commutative algebra this is called an integral domain, while noncommutative conventions vary.
• Indecomposable module
  • … transfers from A canonical instance directly demonstrates that the module is nonzero and every direct-sum decomposition has a zero summand, with ring side and finiteness hypotheses explicit. to An applied instance preserves the same invariant under a changed notation, scale, implementation, or empirical setting..
• Maximal Ideal
  • Therefore \(\mathfrak m\) is maximal exactly when the nonzero quotient \(R/\mathfrak m\) has no nonzero proper two-sided ideals—that is, when the quotient is a simple ring.

    This sourceGraduate Texts in Mathematics 131. Springer, 2001. Authoritative source for two-sided, one-sided, simple-ring, simple-module, Jacobson-radical, local, and semilocal distinctions.

• Nilradical of a ring
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Perfect ring
  • … between constitutive identity and a convenient observable transfers from A canonical example satisfies the full defining rule of Perfect ring with all parameters and conventions explicit. to A careful use of Perfect ring tests the constitutive rule and nearest confusable rather than relying on topical resemblance..
• Primitive ring
  • … module and action, simplicity and faithfulness, annihilator, maximal one-sided ideal characterization, Jacobson density representation and examples separating left from right are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Principal Ideal
  • More generally, “smallest ideal containing (a)” safely includes the additive closure required by the ring’s identity convention.
• Semiprimitive ring
  • … and a convenient observable transfers from The integers are semiprimitive because the intersection of all maximal ideals is zero, although the integers are not semisimple Artinian. to An algebraist distinguishes Jacobson radical zero from absence of every other radical and records whether rings have identity..
• Semisimple module
  • … constitutive identity and a convenient observable transfers from A finite-dimensional representation of a finite group over a characteristic-zero field is semisimple by Maschke's theorem. to An algebraist states ring and finiteness hypotheses and does not infer semisimplicity merely from having a composition series..

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