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\begin{document}

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\section*{radical}

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

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

[[!include higher algebra - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{radical_functors}{Radical functors}\dotfill \pageref*{radical_functors} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

For a \emph{[[commutative ring]]} one defines a \textbf{radical $\sqrt{I}$ of an ideal} $I\subset R$ as an ideal

\begin{displaymath}
\sqrt{I} = \{ r\in R \,|\, \exists n\in \mathbb{N}, r^n\in I \}
\end{displaymath}
An ideal is called a \emph{radical ideal} if it is equal to its own radical.

The [[nilradical|Nilradical]] of a commutative ring is the radical of the $0$ ideal.

For a [[noncommutative ring]] or an [[associative algebra]] there are many competing notions of a radical of a ring such as [[Jacobson radical]], Levitzky radical, and sometimes of radicals of ideals or, more often, of radicals of arbitrary modules of a ring.

\hypertarget{radical_functors}{}\subsection*{{Radical functors}}\label{radical_functors}

Each of the notions of radical mentioned above are functorial, and some of the abstract properties of such functors are used in noncommutative localization theory, when defining so called \emph{radical functors}. Classically these were considered for module categories ${ }_R Mod$ (left modules over a ring $R$, but there are generalizations for arbitrary [[Grothendieck categories]], and there are also some notions of radical for nonadditive categories. See \hyperlink{Shulgeifer60}{Shulgeifer 60}.

We define here radical functors on ${ }_R Mod$, but warn that there are some terminological discrepancies across the literature.

However they are defined, all notions of radical involve [[additive functor|additive]] [[subfunctors]] $i: \sigma \hookrightarrow 1_{ _R Mod}$ of the identity on ${ }_R Mod$, the additive category of left $R$-modules. [[natural transformation|Naturality]] of $i$ implies the equation $i \circ \sigma i = i \circ i\sigma$, whence $\sigma i = i\sigma$ by [[monomorphism|monicity]] of $i$. Some authors refer to these as \emph{preradical functors} (e.g., \hyperlink{Mirhosseinkhani2010}{Mirhosseinkhani 2010}).

Such a functor $\sigma: {}_R Mod\to {}_R Mod$ is \textbf{idempotent} if $\sigma i = i\sigma: \sigma\sigma \to \sigma$ is an isomorphism, and is called a \textbf{radical functor} if in addition $\sigma(M/\sigma(M))=0$ for all $M$ in ${}_R Mod$. Note however that some authors call \emph{this} a \emph{preradical functor}, and define a radical functor to be such a preradical functor that is left exact.

Following \hyperlink{Goldman1969}{Goldman 1969}, a left exact additive subfunctor of the identity is called an \textbf{idempotent kernel functor}. Observe that such is idempotent by the calculation

\begin{displaymath}
\sigma \sigma M = \sigma Ker(M\to M/\sigma M) = Ker (\sigma M\to \sigma(M/\sigma M)) = Ker(\sigma M\to M\to M/\sigma M) = \sigma M
\end{displaymath}
where in the last step, we used that $\sigma$ is a subfunctor of the identity, hence the compositions $\sigma M\hookrightarrow M\to M/\sigma M$ and $\sigma M\to \sigma(M/\sigma M)\to M/\sigma M$ coincide.

However, beware that other authors call a left exact additive subfunctor $\sigma: {}_R Mod\to {}_R Mod$ of the identity functor a [[kernel functor]], and then call a kernel functor $\sigma$ an \emph{idempotent kernel functor} if $\sigma(M/\sigma(M))=0$ for all $M$ in ${}_R Mod$. In other words, their idempotent kernel functors coincide with what other authors call radical functors in the strong (left exact) sense above.

See \hyperlink{BuesoJaraVerschoren95}{Bueso-Jara-Verschoren 95}

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Example (\hyperlink{BuesoJaraVerschoren95}{Bueso-Jara-Verschoren 95 2.3.4}): Let $I$ be a two-sided ideal in a ring. Define a functor $\sigma : {}_R Mod\to {}_R Mod$ on objects by $\sigma M = \{ m\in M\,|\, \exists n, I^n M = 0\}$; it is left exact and idempotent. If $I$ is finitely generated as left $R$-ideal (i.e. as a left $R$-submodule of $R$) then $I$ is a left exact radical functor. It is clear that the formula for $\sigma M$ reminds the definition of the radical of an ideal of a commutative ring.

Nonexample: the subfunctor of identity which to any module $M$ assigns its [[socle]] is left exact but not a radical functor.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item E. G. Shulgefer (. . ), \emph{     }, . ., 51(93):4 (1960), 487--500 \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=sm&paperid=4826&what=fullt&option_lang=rus}{pdf}


\item J. L. Bueso, P. Jara, A. Verschoren, \emph{Compatibility, stability, and sheaves}, Monographs and Textbooks in Pure and Applied Mathematics, 185. Marcel Dekker, Inc., New York, 1995. xiv+265 pp.


\item O. Goldman, \emph{Rings and modules of quotients}, J. Algebra 13 (1969), 10-47.



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Radical_of_a_ring}{Radical of a ring}}

\end{itemize}
[[!redirects radicals]] [[!redirects radical functor]]

[[!redirects radical ideal]] [[!redirects radical ideals]]



\end{document}