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Everything about Division ring

Algebraic structure also called skew field

In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse; that is, an element usually denoted a–1, such that a a–1 = a–1 a = 1. So, (right) division may be defined as a / b = a b–1, but this notation is avoided, as one may have a b–1 ≠ b–1 a. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Division ringA 5-minute orientation
  2. Skim a recent paper: Engineering microbial division of labor for plastic upcyclingInfluential work since 2022
  3. Pick up a classic book: Skew linear groupsThe most-reprinted book on the subject
  4. Read the landmark paper: FtsZ ring structure associated with division in Escherichia coliCited 1,316 times
  5. Go sideways: Noncommutative ringA closely related topic
  6. Go sideways: Associative algebraA closely related topic

📖 Definitions

Noun A ring with 0 ≠ 1, such that every non-zero element a has a multiplicative inverse, meaning an element x with ax = 1.

🗂️ The subject at a glance

  1. Relation to fields and linear algebra
  2. Examples
  3. Main theorems
  4. Related notions

🏆 Landmark papers (most cited)

🆕 Influential since 2022

📚 Books worth reading

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