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\begin{document}

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\section*{nonunital ring}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{for_rings}{For rings}\dotfill \pageref*{for_rings} \linebreak
\noindent\hyperlink{for_algebras}{For $\mathbb{E}_k$-algebras}\dotfill \pageref*{for_algebras} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{unitization}{Unitization}\dotfill \pageref*{unitization} \linebreak
\noindent\hyperlink{AsSlicesOfRings}{Nonunital rings as slices of rings}\dotfill \pageref*{AsSlicesOfRings} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{ReferencesTheory}{Nonunital ring theory}\dotfill \pageref*{ReferencesTheory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of \emph{nonunital ring} is like that of [[ring]] but without the requirement of the existence of an [[identity]] element (``unit'' element).

Nonunital rings with [[homomorphisms]] between them form the [[category]] [[Rng]].

Historically, this was in fact the original meaning of ``ring'', and while mostly ``ring'' has come to mean by default the version with identity element, nonunital rings still play a role (see e.g. the review in \hyperlink{Anderson06}{Anderson 06}) and in some areas of mathematics ``nonunital ring'' is still the default meaning of ``ring''. In particular, non-unital rings may naturally be identified with the [[augmentation ideals]] of $\mathbb{Z}$-[[augmented algebra|augmented]] unital rings, see the discussion \hyperlink{AsSlicesOfRings}{below}.

\begin{remark}
\label{}\hypertarget{}{}
The term ``non-unital ring'' may be regarded as an example of the ``[[red herring principle]]'', as a non-unital ring is not in general a ring in the modern sense of the word.

In [[Bourbaki|Bourbaki 6, chapter 1]] the term \emph{pseudo-ring} is used, but that convention has not become established.

Another terminology that has been suggested for ``nonunital ring'', and which is in use in part of the literature (e.g. \hyperlink{Anderson06}{Anderson 06}) is ``rng'', where dropping the ``i'' in ``ring'' is meant to be alluding to the absence of \emph{i}dentity elements. This terminology appears in print first in (\hyperlink{Jacobson}{Jacobson}), where it is attributed to Louis Rowen.

Similarly there is, for whatever it's worth, the suggestion that a ring without negatives, hence a [[semiring]], should be called a \emph{[[rig]]} (although here one may make a technical distinction about the \emph{additive} identity).

\end{remark}
\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{for_rings}{}\subsubsection*{{For rings}}\label{for_rings}

Specifically:

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{nonunital ring} or \emph{rng} is a [[set]] $R$ with operations of addition and multiplication, such that:

\begin{itemize}%
\item $R$ is a [[semigroup]] under multiplication;
\item $R$ is an [[abelian group]] under addition;
\item multiplication distributes over addition.

\end{itemize}
\end{defn}
More sophisticatedly, we can say that, just as a ring is a [[monoid object]] in [[Ab]], so

\begin{defn}
\label{}\hypertarget{}{}
A \emph{nonunital ring} or \emph{rng} is a [[semigroup]] object in [[Ab]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
A non-unital ring may well contain an element that behaves as the [[identity]] element, i.e. it may be in the image of the [[forgetful functor]] from unital rings to nonunital rings. But if so, then this element is still not part of the defining data and in particular a [[homomorphism]] of non-unital rings need not to preserve whatever identity elements may happen to be present.

\end{remark}
\hypertarget{for_algebras}{}\subsubsection*{{For $\mathbb{E}_k$-algebras}}\label{for_algebras}

[[nonunital Ek-algebras]] are discussed in (\hyperlink{Lurie}{Lurie, section 5.2.3}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{unitization}{}\subsubsection*{{Unitization}}\label{unitization}

\begin{defn}
\label{Unitisation}\hypertarget{Unitisation}{}
Given a non-unital commutative ring $A$, then its \emph{[[unitisation]]} is the [[commutative ring]] $F(A)$ obtained by [[free construction|freely]] adjoining an [[identity]] element, hence the ring whose underlying [[abelian group]] is the [[direct sum]] $\mathbb{Z} \oplus A$ of $A$ with the [[integers]], and whose product operation is defined by

\begin{displaymath}
(n_1, a_1) (n_2, a_2) \coloneqq (n_1 n_2, n_1 a_2 + n_2 a_1 + a_1 a_2)
  \,,
\end{displaymath}
where for $n \in \mathbb{Z}$ and $a \in A$ we set $n a \coloneqq \underset{n\;summands}{\underbrace{a + a + \cdots + a}}$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
In the unitization $\mathbb{Z} \oplus A$ we have $(n,0) + (0,a) = (n,a)$ and hence it makes sense to abbreviate $(n,a)$ simply to $n+a$. The product in the [[unitisation]] is then fixed by the defining requirement that $1 \cdot a = a$ and by the [[distributivity law]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
One can also think of the unitisation as the quotient of the polynomial ring ``$A[x]$'' ($A$ ``with a generic element adjoined'', i.e. a term algebra) quotiented by the relations $ax - a = 0, xa - a = 0$, so that this $x$ must be a right and left identity for multiplication $a$. Cosets must be represented by expressions of the form $a + nx$; this provides an obvious isomorphism to the above definition, and motivates the multiplication for $\mathbb{Z} \oplus A$ above.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Since $A$ embeds into its unitisation $F(A)$, every [[rng]] lives as an ideal in some [[unital ring|ring]]. We can consider $A$-linear actions $A \curvearrowright M$ on abelian groups $M$. Since the endomorphism rings of abelian groups are always unital, the [[universal property]] of the unitisation-forgetful adjunction (see below) ensures that there is a unique extension of the action map $A \to \operatorname{End}_{\mathbf{Ab}}(M)$ along $A \hookrightarrow \operatorname{for} \circ F(A)$ to an action map $F(A) \to \operatorname{End}_{\mathbf{Ab}}(M)$. This induces an equivalence $A\text{-}\mathbf{Mod} \simeq F(A)\text{-}\mathbf{Mod},$ so that one may as well study $F(A)$ if one wanted to study $A$ through its module category.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Similar [[unitisation]] prescriptions work for non-commutative rings and for [[nonunital algebras]] over a fixed base ring, see also at

\begin{itemize}%
\item \emph{[[unitisation of C\emph{-algebras]]\_.}}

\end{itemize}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
Unitisation in def. \ref{Unitisation} extends to a [[functor]] from $CRng$ to [[CRing]] which is [[left adjoint]] to the [[forgetful functor]] from [[commutative rings]] to non-unital commutative rings.

\begin{displaymath}
F \colon CRng \leftrightarrow CRing \colon U
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This is because the definition of any ring [[homomorphism]] out of $F(A)= (\mathbb{Z} \oplus A, \cdot)$ is uniquely fixed on the $\mathbb{Z}$-summand.

\end{proof}
\hypertarget{AsSlicesOfRings}{}\subsubsection*{{Nonunital rings as slices of rings}}\label{AsSlicesOfRings}

\begin{defn}
\label{AugmentationIdealFunctor}\hypertarget{AugmentationIdealFunctor}{}
Write $CRing_{/\mathbb{Z}}$ for the [[slice category]] of [[CRing]] over the ring of [[integers]] ([[augmented algebra|augmented commutative rings]]). Write

\begin{displaymath}
CRing_{/\mathbb{Z}} \longrightarrow CRng
\end{displaymath}
for the [[functor]] to commutative non-unital rings which sends any $(R \stackrel{\phi}{\to} \mathbb{Z})$ to its \emph{[[augmentation ideal]]}, hence to the [[kernel]] of $\phi$

\begin{displaymath}
(R \stackrel{\phi}{\to} \mathbb{Z}) \mapsto ker(\phi)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{EquivalenceOfNonunitalAndSliced}\hypertarget{EquivalenceOfNonunitalAndSliced}{}
The [[augmentation ideal]]-functor in def. \ref{AugmentationIdealFunctor} is an [[equivalence of categories]] whose inverse is given by [[unitisation]], def. \ref{Unitisation}, remembering the [[projection]] $(\mathbb{Z} \oplus A) \to \mathbb{Z}$:

\begin{displaymath}
CRng \simeq CRing_{/\mathbb{Z}}
  \,.
\end{displaymath}
\end{defn}
\begin{proof}
That the functor is [[fully faithful]] is to observe that for a ring $R \stackrel{\phi}{\to} \mathbb{Z}$ the [[fiber]] $R_{n}$ over $n \in \mathbb{Z}$ is a [[torsor]] over the additive group underlying the [[augmentation ideal]] $A = R_0 = ker(\phi)$, and moreover it is a pointed torsor, the point being $n$ itself, hence is canonically equivalent to the [[augmentation ideal]] $A$, the equivalence being addition by $n$ in $R$. Hence any homomrphism of rings with identity over $\mathbb{Z}$

\begin{displaymath}
\itexarray{
    R_1 && \stackrel{f}{\longrightarrow} && R_2
    \\
    & {}_{\mathllap{\phi_1}}\searrow && \swarrow_{\mathrlap{\phi_2}}
    \\
    && \mathbb{Z}
  }
\end{displaymath}
is uniquely fixed by its restriction to the augmentation ideal $ker(\phi_1)$, whose image, moreover, has to be in the augmentation ideal $ker(\phi_2)$.

\end{proof}
The identification of non-unital algebras as augmentation ideals of augmented unital algebras is used for instance in (\hyperlink{Fresse06}{Fresse 06}).

\begin{remark}
\label{}\hypertarget{}{}
In terms of [[arithmetic geometry]], the [[Isbell duality|formally dual]] statement of prop. \ref{AugmentationIdealFunctor} is that arithmetic geometry induced by non-unital rings is equivalently ordinary arithmetic geometry \emph{under} [[Spec(Z)]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The generalization of prop. \ref{EquivalenceOfNonunitalAndSliced} to [[nonunital Ek-algebras]] is (\hyperlink{Lurie}{Lurie, prop. 5.2.3.15}).

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[unitisation of C\emph{-algebras]]}


\item [[noncommutative ring]], [[nonassociative ring]]


\item [[rig]]


\item [[nonunital Ek-algebra]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{ReferencesTheory}{}\subsubsection*{{Nonunital ring theory}}\label{ReferencesTheory}

The terminology ``rng'' originates in

\begin{itemize}%
\item Nathan Jacobson \emph{Basic Algebra}.

\end{itemize}
A survey of commutative rng theory is in

\begin{itemize}%
\item D. Anderson, \emph{Commutative rngs}, in J. Brewer et al. (eds.) \emph{Multiplicative ideal theory in Commutative Algebra}, 2006

\end{itemize}
Discussion of [[module]] theory over rngs (with close relation to [[torsion modules]]) is in

\begin{itemize}%
\item [[Daniel Quillen]], \emph{Module theory over nonunital rings}, August 1996 ([[QuillenModulesOverRngs.pdf:file]])

\end{itemize}
Discussion of non-unital rings as [[augmentation ideals]] of augmented unital rings includes

\begin{itemize}%
\item [[Benoit Fresse]], \emph{The Bar Complex of an E-infinity Algebra}, Adv. Math. 223 (2010), pages 2049-2096 (\href{http://arxiv.org/abs/math/0601085}{arXiv:math/0601085})

\end{itemize}
A definition of [[algebraic K-theory]] for nonunital rings is due to

\begin{itemize}%
\item [[Daniel Quillen]], \emph{$K_0$ for nonunital rings and Morita invariance}, J. Reine Angew. Math., 472:197-217, 1996.

\end{itemize}
with further developments (in [[KK-theory]]) including

\begin{itemize}%
\item [[Snigdhayan Mahanta]], \emph{Higher nonunital Quillen K'-theory, KK-dualities and applications to topological T-dualities}, J. Geom. Phys., 61 (5), 875-889, 2011. (\href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf}{pdf})

\end{itemize}
Discussion in the context of [[higher algebra]] ([[nonunital Ek-algebras]]) is in

\begin{itemize}%
\item [[Jacob Lurie]], section 5.2.3 of \emph{[[Higher Algebra]]}

\end{itemize}
[[!redirects rng]] [[!redirects rngs]]

[[!redirects nonunital ring]] [[!redirects nonunital rings]] [[!redirects non-unital ring]] [[!redirects non-unital rings]]

[[!redirects unitization of a ring]] [[!redirects unitization of rings]] [[!redirects unitizations of rings]] [[!redirects unitisation of a ring]] [[!redirects unitisation of rings]] [[!redirects unitisations of rings]]

[[!redirects unitization of a commutative ring]] [[!redirects unitization of commutative rings]] [[!redirects unitizations of commutative rings]] [[!redirects unitisation of a commutative ring]] [[!redirects unitisation of commutative rings]] [[!redirects unitisations of commutative rings]]



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