Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
Questions tagged [ra.rings-and-algebras]
3185 questions
47
votes
2 answers
Non isomorphic finite rings with isomorphic additive and multiplicative structure
About a year ago, a colleague asked me the following question:
Suppose $(R,+,\cdot)$ and $(S,\oplus,\odot)$ are two rings such that $(R,+)$ is isomorphic, as an abelian group, to $(S,\oplus)$, and $(R,\cdot)$ is isomorphic (as a semigroup/monoid)…
Arturo Magidin
- 7,012
47
votes
9 answers
What are the reasons for considering rings without identity?
I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.
teil
- 4,261
39
votes
5 answers
When does a ring surjection imply a surjection of the group of units?
The following might be a very trivial question. If so, I don't mind it being closed, but would appreciate a reference where I could read about it.
Let $R$ and $S$ be commutative rings and let $R^\times$ and $S^\times$ denote their respective…
José Figueroa-O'Farrill
- 32,811
32
votes
7 answers
Consequences of not requiring ring homomorphisms to be unital?
As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1_R)=1_S$. But there has been a minority who do not require this, one prominent example being Herstein in Topics in Algebra.
What are some of…
Zev Chonoles
- 6,722
28
votes
1 answer
Formally real Jordan algebras
In 1934, Jordan, von Neumann and Wigner gave a nice classification of finite-dimensional simple Jordan algebras that are 'formally real', meaning that a sum of squares is zero only if each term in the sum is zero. In 1983 Zelmanov generalized this…
John Baez
- 21,373
24
votes
14 answers
Ring with three binary operations
A rather precocious student studying abstract algebra with me asked the following question: Are there interesting rings where there are not just two but three binary operations along with some appropriate distributivity properties?
Deane Yang
- 26,941
19
votes
1 answer
$\Lambda$-Ring Structures on $\mathbb A^2$
A $\Lambda$-ring structure on a torsion-free ring over $\mathbb Z$ is a commuting family of endomorphisms $\psi_p$ satisfying $\psi_p(x) \equiv x^p$ mod $p$.
One $\Lambda$-ring structure on $\mathbb Z[x]$ is defined by $\psi_p(x)=x^p$.
Another can…
Will Sawin
- 135,926
18
votes
3 answers
Finite non-commutative ring with few invertible (unit) elements
for a ring $R$ with unity , let $U(R)$ denote the group of units of $R$ . Now there are lots of finite commutative rings, of arbitrarily high order, with exactly one unit ; indeed $U(R)=1$ for a finite commutative ring $R$ iff $a^2=a , \forall a…
user111524
18
votes
3 answers
Are there countable index subrings of the reals?
Does ${\mathbb R}$ have proper, countable index subrings? By countable I mean finite or countably infinite. By subring I mean any additive subgroup which is closed under multiplication (I don't care if it contains $1$.) By index, I mean index as…
Fabrizio Polo
- 671
16
votes
3 answers
Two infinite dimensional algebras such that the center of their tensor product is bigger than the tensor product of their centers
I am searching for two infinite dimensional algebras such that the center of their tensor product is bigger than the tensor product of their centers. Who knows of such examples? Thanks a lot.
user37656
- 161
16
votes
1 answer
Free modules over integers
After the course of linear algebra I'm more familiar with vector spaces rather than modules so my question may seem to be silly but I think it's quite natural for someone who thinks of
modules as 'vector spaces over a ring': which of the following…
truebaran
- 9,140
15
votes
2 answers
Is every finitely generated idempotent ring singly generated as a two-sided ideal?
In this post, a ring is understood to be what one usually calls a ring, not assuming that it has a unit. Some people call such objects rng.
Question: Let R be a finitely generated (non-unital and associative) ring, such that $R=R^2$, i.e. the…
Andreas Thom
- 25,252
15
votes
1 answer
Non-isomorphic algebras becoming isomorphic after adding identities: mistake in an exercise
In the book: Drozd, Y.A., Kirichenko, V.V.: Finite dimensional algebras. Springer, Berlin (1994), there is an exercise which suggests a positive answer to the next question, and Ryszard R. Andruszkiewicz gives a counterexample:
Let $A,B$ be two…
Or Shahar
- 421
14
votes
3 answers
Polynomial Rings
Let $R$ and $S$ be non-zero rings with identity. Is it possible to have $R[x] \cong S[[x]]$ ?
user30230
12
votes
4 answers
Subalgebras of matrices
Given a matrix algebra over a field, can one describe all its subalgebras?
chana
- 121