The relationship of ${\frak m+m=m}$ to AC Two simple questions: (Of course ${\frak m}$ denotes a cardinal in the weak sense in the claims below.)


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*Can we prove in ZF that $\aleph_0\le{\frak m\Rightarrow m+m=m}$?

*If not, what is the relationship of this statement for all cardinals ${\frak m}$ to AC?


The weaker principle $\aleph_0\le{\frak m\Rightarrow m}+1={\frak m}$ can be derived in ZF, and it is a well-known result of Tarski that $\forall{\frak m}\ge\aleph_0,{\frak m\times m=m}$ implies the axiom of choice, but I am not sure about the intermediate result with ${\frak m+m=m}$.
Edit: A related question: is it consistent with ZF that there exists a cardinal ${\frak m}\ge\aleph_0$ such that ${\frak m=n+p}$ with ${\frak n<m}$ and ${\frak p<m}$? It's hard to say whether ${\frak m+m=m}$ precludes this scenario.
 A: Suppose that $A$ is infinite and Dedekind finite. Then $\mathfrak m=|A\cup\mathbb N|$ satisfies that $|A|<\mathfrak m$, $\aleph_0<\mathfrak m$, and $\mathfrak m+\mathfrak m>\mathfrak m$. 
To see the last inequality, note that if $\mathfrak m+\mathfrak m=\mathfrak m$ then $A\times 2$ embeds into $A\cup\mathbb N$, say via $f$, but only a finite subset of it embeds into $\mathbb N$, so a set strictly larger than $A$ must embed into $A$. 
Note that being Dedekind infinite is the same as embedding $\omega$, so if we require $\mathfrak m+\mathfrak m=\mathfrak m$ for all infinite cardinals $\mathfrak m$, or even for all Dedekind-infinite cardinals, then there are no Dedekind-finite sets.  But no, Countable Choice is strictly stronger than the lack of infinite Dedekind finite sets. This is due to Pincus, see this MO question. 
As for whether $\mathfrak m+\mathfrak m=\mathfrak m$ (the idemmultiple hypothesis) gives us Countable Choice, the answer is again no, as proved by 

Gershon Sageev. An independence result concerning the axiom of choice, Ann. Math. Logic 8, (1975), 1–184. MR0366668 (51 #2915).

A: To add slightly on Andres' answer, it is not only that $\frak m+m=m$ does not imply $\sf AC_\omega$, but in the other direction we can have $\sf DC_\kappa$ (for an arbitrary $\kappa$) but $\frak m+m=m$ fails.
We say that $A$ is a $\kappa$-amorphous set if every subset of $A$ has cardinality $<\kappa$, or its complement has such cardinality - but not both.
Note that if $A$ is $\lambda$-amorphous, then for $\kappa>\lambda$, $A$ is $\kappa$-amorphous as well. We say that $A$ is a properly $\kappa$-amorphous set if it is $\kappa$-amorphous, and $\aleph(A)=\kappa$. In other words, if every ordinal smaller than $\kappa$ can be injected into $A$.
For every regular $\kappa$ it is consistent with $\sf ZF+DC_{<\kappa}$ that there exists a [properly] $\kappa$-amorphous set. 
Now, if $A$ is a $\kappa$-amorphous set, then $|A|<|A|+|A|$. This is a trivial observation since $A\times2$ can be written as the union of two sets, neither has complement of size $<\kappa$. So there cannot be an injection from $A\times2$ into $A$, but there is an obvious injection in the other direction.
Therefore $\sf ZF+DC_\kappa$ cannot prove $\frak\forall m. m+m=m$.
(On a slightly more confusing note, I am switching between $\sf DC_\kappa$ and $\sf DC_{<\kappa}$, but the former can be thought as $\sf DC_{<\kappa^+}$.)

To the added question, by the way, this is a simple result. If I recall correctly it is by Tarski.

The axiom of choice is equivalent to the assertion that whenever $\frak p,m,n$ are cardinals and $\frak p+m=n$ then $\frak p=n$ or $\frak m=n$.

The proof is simple, let $\frak a$ be a cardinal, and let $\kappa=\aleph(\frak a)$ the Hartogs number of a set of size $\frak a$. Then $\frak a+\kappa=a$ or $\frak a+\kappa=\kappa$. But $\kappa\nleq\frak a$ so the first option is impossible, therefore $\frak a\leq\kappa$, and can be well-ordered. $\square$
In particular if the axiom of choice fails, let $\frak a$ be a non-well ordered cardinal and $\kappa$ its Hartogs number, then $\frak a+\kappa$ is a set which can be decomposed into two strictly smaller cardinals.
It is also consistent with $\sf ZF$ that every cardinal which is not well-orderdable can be written in such way:

Monro, G.P. Decomposable Cardinals. Fund. Math. vol. 80 (1973), no. 2, 101–104. 

