Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Misunderstanding on Invariant Basis Number

Jean-Baptiste Vienney (Jun 14 2024 at 21:55):

The following is driving me absolutely crazy. I'm misunderstanding something but I don't find what it is.

Let $R$ be a ring. In these notes (corrected the link), they say that:
(1) if $I$ is a proper ideal of $R$, and if $R/I$ has the invariant basis number (IBN), then $R$ also has the IBN,
(2) every division ring has the IBN.

With my understanding, it would follow that every nonzero ring has the IBN (which is false).

Here is my reasoning: Let $R$ be a nonzero ring. Let $I$ be a maximal ideal of $R$, which exists by Krull's theorem. Then $R/I$ is a division ring, therefore $R/I$ has the IBN. Hence $R$ has the IBN.

Where is the mistake??

Todd Trimble (Jun 14 2024 at 22:05):

It's probably in assuming that a ring modulo a maximal two-sided ideal is a division ring, as opposed to a simple ring.

For example, the maximal two-sided ideal of an $n \times n$ matrix algebra is the zero ideal.

Jean-Baptiste Vienney (Jun 14 2024 at 22:12):

Oh thanks, it should be that. They say here that the quotient of a ring $R$ by a two-sided ideal $I$ is a division ring iff $I$ is maximal as a left ideal or as a right ideal.