# Questions tagged [abstract-algebra]

Deprecated; do NOT use this tag. Instead you could consider gr.group-theory, ac.commutative-algebra, ra.rings-and-algebras, universal-algebra, or various more specific tags.

314 questions
86 views

### Do these sorts of submonoids go by a particular name?

Given any monoid $M$ for every element $x\in M$ we can define two submonoids of $M$ as follows: $$r(x)=\{y\in M:xy=x\}$$ $$l(x)=\{y\in M:yx=x\}$$ Do these sorts of sub-monoids go by a particular name?...
78 views

304 views

83 views

### Isomorphism concerning $Soc(M_n(R))$

It is known that $M_n(R/J(R))\simeq M_n(R)/M_n(J(R))=M_n(R)/J(M_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R_R)$, where $J(R)$ and $Soc(R_R)$ stand for the Jacobson ...
213 views

### Does the tensor algebra $T(V)$ of $V$ isomorphic to the symmetric algebra of the free Lie algebra over $V$?

Let $V$ be a finite dimensional vector space. Let $T(V)$ be the tensor algebra over $V$. Do we have $T(V) \cong S(Lie(V))$ as a graded vector space? Here $S(Lie(V))$ is the symmetric algebra of the ...
493 views

### Has the Jacobson/ Baer radical of a group been studied?

On groupprops, the Jacobson or Baer radical of a group $G$ is defined to be the intersection of all maximal normal subgroups of $G$. This is similar to, but distinct from, the Frattini subgroup which ...
215 views

81 views

74 views

### Extension field $\mathbb{C}(t,u)$ over $\mathbb{C}(t^n,u^n)$
Let $R=k[x_1,\ldots,x_6]$ be a polynomial ring and $I=(x_1x_5-x_2x_4,x_2x_6-x_3x_5)$ be an ideal. How to show that, $(I^2:x_1x_5-x_2x_4)=I$ ?