Can you expand an abelian group to have elements that satisfy a specific equation?

Question

Given an abelian group $G$ and an equation of the form $$n_1 a_1 + ... + n_r a_r = m_1 y_1 + ... + m_k y_k$$where the $n_i$ and $m_i$ are positive integers and the $a_i$ are elements of $G$, is there guaranteed to be an abelian group containing $G$ as a subgroup such that there are $y_i$ satisfying the equation?

Answer 1

Yes. It suffices to show that, for any $m>0$ and $a\in G$, there exists a injection of $\mathbb{Z}$-modules $\alpha:G\to H$ such that $\alpha(a)=my$ for some $y\in H$. (Why?) Thus let $H$ be the quotient of $G\times\mathbb{Z}$ by the subgroup $L\leqslant G$ generated by $(a,-m)$. We claim that the natural map $\alpha:G\to H$ taking $x$ to $(x,0)+L$ is an injection. Indeed, suppose $(x,0)+L$ is zero. Then $(x,0)\in L$, so $(x,0)=k(a,-m)=(ka,-km)$ for some $k\in\mathbb{Z}$. But, since $m>0$, we have $km\neq 0$ for every $k\in\mathbb{Z}\setminus\{0\}$, so this forces $k=0$ and thus $(x,0)=(0,0)$ and thus $x=0$, as desired. So indeed we may consider $G$ as a subgroup of $H$. But note that $$\alpha(a)=(a,0)+L=(a,0)-(a,-m)+L=(0,m)+L=m\big((0,1)+L\big),$$ so taking $y=(0,1)+L$ gives the desired solution.

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The idea here is that, for a ring $R$, an index set $I$, and an $R$-module $M$, you can often use a suitable quotient of $M\times R^{\oplus I}$ to add $I$-many new elements to $M$ with properties you desired. (Some properties are not realizable in this way – for instance, we of course could not have taken $m=0$ in the above equation if $a\neq 0$ – but it's a useful slogan.) As an aside, since you tagged with model-theory, you may want to look into pure-injective modules, which deal precisely with solutions sets of the kind of equations you are considering. (The existence of simultaneous solutions to these kinds of equation is first-order expressable in the language $$\mathcal{L}_{R\text{-mod}}=(+,-,0,r)_{r\in R}$$ of $R$-modules, where each $r\in R$ is considered as a unary function symbol. These formulas are called "positive-primitive" formulas, and every $\mathcal{L}_{R\text{-mod}}$-formula is in fact, modulo the complete theory of any $R$-module, equivalent to a Boolean combination of positive primitive formulas.) A classic paper on the subject is Ziegler's.