characterization of a submodule - MathOverflow

characterization of a submodule

2010-02-11T08:17:27Z

In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if it is closed under addition and scalar multiplication. For a module $M$ over a ring $R$ with identity the similar result is true. Is it true in modules over a nonunital ring?

I should mention that the analogue of the following is not true.

In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if for all scalars $\alpha_1, \alpha_2$ and for all $w_1, w_2 \in W$, $\alpha_1 w_1 + \alpha_2 w_2 \in W$.

This can be seen by the module $\mathbb{Z}$ over $2\mathbb{Z}$ and the subset $2\mathbb{Z} \cup \{-3,3\}$.

Answer by Shizhuo Zhang for characterization of a submodule

2010-02-11T11:12:14Z

First, Let me talk about the "correct definition" of module over non-unital ring(not necessarily commutative) and how this definition coincide with usual definition of module over unital ring in particular case

First we study $R-mod_{1}$={category of associative action of $R$ on $k$-mod}= {($M$,$R\bigotimes _{k}M\rightarrow M$).

$r_{1}(r_{2}z)=(r_{1}r_{2})z$}

Let $R_{1}=R\bigoplus k$ be an untial $k$-algebra with usual multiplication. And we have the categorical equivalence as: $R-mod_{1}\approx R_{1}-mod$

Now,we define module over non-unital algebra $R$ as $R-mod=R_{1}-mod/(Tors_{R_{1}})^{-}$, where $(Tors_{R_{1}})^{-}$ is Serre subcategory of $R_{1}-mod$

$R_{1}-mod\overset{q_{R}^{*}}{\rightarrow}R-mod$ is a localization functor having right adjoint functor.

Trivial Example:

if $R$ has is an unital $k$-algebra. Then $R_{1}-mod$ is equivalent to $R-mod$

Less Trivial example in commutative case:

Consider affine line $k[x]$. Let $R=xk[x]$(maximai ideal of $k[x]$). Then $R-mod$=$Qcoh(\mathbb{A}^{1}-{0}$). It is a cone.

Toy general case:

Let $m$ is a two-sided proper ideal of associative commutative unital ring $A$. Then: we have

$m-mod$=$A-mod/({M\epsilon A-mod|m\cdot M=0})^{-}$, where$T^{-}$ is smallest Serre category containing $T$. It is clear that is equivalent to Qcoh(Complement of $\mathbb{V}(m)$),where $\mathbb{V}(m)$ is closed subvariety determined by $m$.

Now, I should stop here and write another(maybe)post on definition of sub-module. There are several reference:

Gabriel, Pierre Des catégories abéliennes. (French) Bull. Soc. Math. France 90 1962 323--448 Kontsevich-Rosenberg Noncommutative spaces and flat descent

Gabber-RameroAlmost Ring Theory

Answer by unknown (google) for characterization of a submodule

2010-02-11T13:45:54Z

In Hungerford everything is defined without assuming that rings have identity (or that they are commutative). At least according to Definition IV.1.3 on p. 171 of Hungerford, the answer to your first question is affirmative. Quoting:

Definition 1.3. Let $R$ be a ring, $A$ an $R$-module and $B$ a nonempty subset of $A$. $B$ is a submodule of $A$ provided that $B$ is an additive subgroup of $A$ and $rb\in B$ for all $r\in R$, $b\in B$.

[I hope that my interpretation of the question was correct. I understood 'characterization of submodule' as 'definition of submodule.']

EDIT: Google books links to the definition of a module, a submodule, and a ring in Hungerford:

http://books.google.com/books?id=t6N_tOQhafoC&lpg=PP1&dq=hungerford&hl=iw&pg=PA169#v=onepage&q=&f=false

http://books.google.com/books?id=t6N_tOQhafoC&lpg=PP1&dq=hungerford&hl=iw&pg=PA171#v=onepage&q=&f=false

http://books.google.com/books?id=t6N_tOQhafoC&lpg=PP1&dq=hungerford&hl=iw&pg=PA115#v=onepage&q=&f=false