\magnification=\magstep1 \hsize=5.5truein \vsize=8.5truein \hoffset=.5truein \baselineskip=15pt plus 2pt minus 2pt %\nopagenumbers %\baselineskip=15pt plus 2pt minus 2pt %\hoffset=30pt \voffset=25pt \vsize=465pt \topskip=20pt %\baselineskip = 20pt plus 1pt minus 1pt %\headline={\hfill\tenrm\folio} \footline={} \font\sm=cmbx7 \font\foot=cmr9 %\def\ssm{\bf \sm} \def\ub{\underbar} \def\sk{\vskip.3truein} \def\Theorem #1 {\vskip20pt {#1} \ {\ub {Theorem}. \ \ } } %\def\Theorem #1 {\vskip20pt\noindent #1 \ {\ub {T}{\sm \ub {HEOREM}: \ \ }}} \def\Proof {\vskip20pt {\ub {Proof}. \ } \rm } %\def\Proof {\vskip20pt \bf {PROOF:} \ \ \rm } \def\Proposition #1 {\vskip20pt {#1} \ {\ub {Proposition}: \ \ } } \def\Lemma #1 {\vskip20pt {#1} \ {\ub {Lemma}: \ \ } } \def\Claim #1 {\vskip20pt \noindent \bf {#1} \ {\bf {CLAIM}: \ \ }} \def\Corollary #1 {\vskip20pt {#1} \ {\ub {Corollary}: \ \ } } \def\Example #1 {\vskip20pt {#1} \ {\ub {Example}: \ \ }} \def\Examples #1 {\vskip20pt \noindent \bf {#1} \ {\bf {EXAMPLES}: \ \ }\rm} \def\Definition #1 {\vskip20pt \noindent \bf {#1} \ {\bf {DEFINITION}: \ \ }\rm} \def\br{\break} \def\brr{\br\br} \def\ds{\displaystyle} \def\qed{{\hfill\vrule height 5pt width 5pt depth 0pt}\vskip10pt} \def\cdd{\cdot\cdot\cdot} \def\qh{quasi-hereditary } \def\adm{$c_1 < \cdd < c_k$ and $c_{[k+1]} \leq k$} \def\admm{$c_n < \cdd < c_{k-1}$, and $c_{k} \leq k$} \def\iq1{$c_{[i+1]} \leq c_i +1$ \ for $1\leq i\leq n$} \def\thb{\overline {\theta}} \def\ss{\medskip \noindent} \outer\def\beginchapter#1. #2\par{\vfill\eject\vglue .25in\centerline {CHAPTER #1} \vskip 10pt\centerline{#2}\message{#2}\vskip10pt} \outer\def\beginsection #1\par{\vskip0pt plus.1\vsize\penalty-150 \vskip0pt plus-.1\vsize\bigskip\vskip\parskip\message{#1} \centerline {{#1}}\nobreak\vskip10pt} %old macro %\outer\def\beginsection #1\par{\vskip0pt plus.1\vsize\penalty-150 \vskip0pt %plus-.1\vsize\medskip\vskip\parskip\message{#1} %{\underbar{#1}}\nobreak\medskip} %\outer\def\beginsection #1\par{\vbox {\vskip10pt {\centerline {\underbar{#1}}} \vskip10pt}} \outer\def\beginsubsection #1\par{\vskip0pt plus.1\vsize\penalty-100 \vskip0pt plus-.1\vsize\vskip20pt\vskip\parskip\message{#1} \centerline{{#1}}\nobreak\medskip} \outer\def\beginsubsubsection #1\par{\vskip0pt plus.1\vsize\penalty-20\vskip0pt plus-.1\vsize\vskip20pt\vskip\parskip\leftline{\underbar{#1}} \nobreak\vskip20pt} \def\det #1 {\mathop{\hbox {\rm det} C(#1) }} \def\gd #1 {\mathop {\hbox {\rm lgldim} #1}} \def\Tr #1 #2 {\mathop {\hbox {\rm Tr}_{#1} ({#2}) }} \def\pd #1 {\mathop {\hbox {\rm pdim} #1}} \def\looo {$\bf ^\alpha \ \hat {}$ \hskip-5.8pt $ \bigcirc$} \def\loo {$\bf \hat {}$ \hskip-5.8pt $\bigcirc$} \def\ann #1 #2 {\mathop {\hbox {\rm lann}_{#1} ({#2})}} \def\ut#1{{\undertext{#1}}} \def\undertext#1{$\underline{\smash{\hbox{#1}}}$} \def\Ker{\mathop {\rm Ker}\nolimits} \def\soc #1 #2 {\mathop {\hbox {\rm Soc} (_{#1} {#2})}} \def\LL #1 {\mathop {\hbox {\rm L} ({#1})}} %franks macros %\headline={\ifnum\pageno<2{} \else % \vbox{\noindent Definition\hfill %{page %\folio}\par\noindent \hrule}\fi} %\parskip=6pt \def\satz#1#2{\vskip24pt\vbox{\item{}{{\bf #1} {\it #2}}}} %begin text %\vbox{\vskip1.2truein} \centerline {\bf {PRIMITIVE HEREDITY IDEALS}} \centerline {Darren D. Wick} \rm \vskip2truecm %\centerline {Darren D. Wick} %\centerline {Millsaps College} %\centerline {Jackson, Mississippi 39210} %\vskip2truecm %\midinsert %\narrower %\narrower %\noindent \bf {Abstract}. \rm Let $R$ be a left artinian ring. Dlab and Ringel have shown that $R$ is hereditary if and only if every chain of idempotent ideals can be refined to an heredity chain (see [1] Theorem 1). In particular, if $R$ is a basic hereditary ring, then every primitive ideal is an heredity ideal. The converse to this is clearly false. (See Example 1). We will introduce a class of rings, that includes serial rings and monomial algebras, for which the converse does hold. %\endinsert \vskip15pt Throughout this paper, $R$ will be a basic left artinian ring (with unity) with a basic set of primitive idempotents $\tau =\lbrace e_1, \cdd ,e_n \rbrace$ and with $J$ the Jacobson radical of $R$. We will refer to the ideals $Re_i R$ as primitive ideals. If $M$ is an $R$-module, we denote the Loewy length of $M$ by $L(M)$, and the composition length of $M$ by $c(M)$. Recall that an ideal $I$ of $R$ is {\it heredity} if $I^2=I$, ${_R}I$ is projective, and $IJI=0$. The ring $R$ is {\it quasi-hereditary} if there exists a chain of ideals, $$0=I_0\subset I_1 \subset \cdots \subset I_l=R,$$ called an {\it heredity chain}, such that $I_k/I_{k-1}$ is an heredity ideal in $R/I_{k-1}$ for each $k=1, \cdots, l$. (see [2] or [1]). If $I$ is an idempotent ideal of $R$, then $I=ReR$ for some idempotent $e$ in $R$. Moreover, $e$ may be taken to have the form $e=e_1 +\cdd +e_k$ for some $k\leq n$ and a suitable ordering of $\tau$ (see [2] Lemma 1.1). Furthermore, if $I=R(e_1 +\cdd + e_k)R$ is an heredity ideal, then each ideal $Re_i R, \ i=1,\cdd ,k$, is again an heredity ideal of $R$ (see [2] Lemma 1.1). Thus (see [2] Corollary 1.2) with a suitable ordering of $\tau$, any heredity chain can be refined to an heredity chain of the form $$0\subset Re_1 R\subset R(e_1 +e_2 )R\subset \cdd \subset R(e_1 + \cdd + e_{n-1} )R \subset R.$$ Dlab and Ringel have shown that $R$ is hereditary if and only if every chain of idempotent ideals can be refined to an heredity chain (see [1] Theorem 1). We first consider the trace of projective left modules in the radical of the ring $R$. If $Re$ is projective, then $ReJ=\Tr J Re $. Note that $ReJ$ is the right radical of the idempotent ideal $ReR$. \vskip10pt \ub {Lemma 1}. Let $ReR$ be a primitive ideal. The following are equivalent. \item {(a)} $ReR$ is an heredity ideal. \item {(b)} $ReJ$ is a projective left $R$-module. \item {(c)} For each $i=1,\cdd ,n$, $\Tr Je_i Re $ is either zero or isomorphic to a direct sum of copies of $Re$. \vskip10pt \ub {Proof}. We may assume $e=e_1$. For any module $_R N$, $Re_1 N=\Tr N Re_1 $. Thus the equivalence of (b) and (c) follows immediately from the direct sum decomposition $$Re_1 J=\oplus_{i=1}^n Re_1 Je_i =\oplus_{i=1}^n \Tr Je_i Re_1 .$$ Since $R$ is basic, $Re_1 Re_i =Re_1 Je_i $ for $i=2,\cdd ,n$. Hence we have the decomposition $$\eqalign {Re_1 R & =\oplus_{i=1}^n Re_1 Re_i \cr & =Re_1 \oplus (\oplus_{i=2}^n Re_1 Je_i) \cr & =Re_1 \oplus (\oplus_{i=2}^n \Tr Je_i Re_1) .}$$ Assume condition (c). From this last decomposition we have that $Re_1 R$ is a direct sum of copies of $Re_1$ and is therefore projective. If $Re_1 Je_1 \not= 0$, then $1\leq c(Re_1 Je_1 )\leq c(Je_1 )1:$ We first note that $x\notin Y_{L(R) -1}$. Since $x=e_i x$ for some basic idempotent $e_i$, $Rx$ is a local left ideal of $R$. Thus if $ReRx=Rx$ we are done. Suppose $ReRx\not= Rx$. Then $ReRx\subset Jx$ and therefore $ReRx=ReJx$. But $Jx=\oplus_{y\in Y_{(l+1)x}} Ry$ by Proposition 1 For $y\in Y_{(l+1)x}$, $Ry \subset Jx$ so that $\LL Ry <\LL Rx $. Thus by induction $ReRy$ is either zero or a direct sum of local left ideals. Thus $ReRx=ReJx=\oplus_{y\in Y_{(l+1)x}} ReRy$ is either zero or a direct sum of local left ideals. Let $1\leq k\leq \LL R -1$. Then by Proposition 1, $J^k =\oplus_{x\in Y_{k}} Rx$. Thus $ReJ^k =\oplus_{x\in Y_{k}} ReRx$ is either zero or a direct sum of local left ideals. \vskip10pt Now, the existence of a tree subset for the regular module $_R R$ is a defining property of left monomial rings (see [3] Definition 2.2). We thus have the following examples of rings satisfying the conditions of Theorem 1 (see [3] p. 6). \vskip10pt \goodbreak \ub {Example 2}. The following rings satisfy the conditions of Theorem 1. \item {(a)} Left monomial rings. \item {(b)} Left serial rings. \item {(c)} Monomial algebras. \item {(d)} Left hereditary left artinian rings. \item {(e)} Left artinian rings with $J^2 =0$. \vskip10pt Recall that a ring $R$ is \it l-hereditary \rm if for every two indecomposable projective $R$-modules $P$ and $Q$, every non-zero homomorphism $h:P \rightarrow Q$ is monic (see [4]). We note that l-hereditary rings are quasi-hereditary (see [2] Proposition 1.3). % This fact relies on the existence of a %simple projective module which generates an heredity ideal in an l-hereditary ring. %However, it is easy to see that not every indecomposable projective module $Re_i$ %in an l-hereditary ring $R$ generates a projective trace ideal $Re_i R$. As was shown by Burgess and Fuller, if $ReR$ is a primitive heredity ideal, then every non-zero homomorphism $h:Re \longrightarrow R$ is monic ([2] Lemma 1.7). Hence if every primitive ideal is heredity, $R$ is l-hereditary. Conversely, suppose that $R$ is l-hereditary and $ReR$ a primitive ideal. Then any non-zero homomorphism $h:Re\longrightarrow Re$ is monic. Thus $eJe \subset ReJe = \Tr Je Re =0$. However, it is easy to see that not every primitive ideal in an l-hereditary ring is an heredity ideal. Consider the following example. \vskip10pt \ub {Example 3}. Let $\Gamma$ be the digraph \vskip10pt \vbox{\settabs 7\columns \+&&& \hskip8.8pt $\buildrel a \over \longrightarrow$ & $\buildrel c \over \longrightarrow$ \cr \vskip-10pt \+&&& 1 & \hskip-15pt 2 & \hskip-25pt 3 \cr \vskip-10pt \+&&& \hskip8.8pt $\buildrel b \over \longrightarrow$ & $\buildrel d \over \longrightarrow$ \cr \+&&&&& \cr} Let $k$ be a field and let $R=k\Gamma /I$ where $I=(cb-da)$. Then the indecomposable projective left $R$-modules have diagrams \goodbreak \vskip12pt \vbox{{\settabs 16\columns \+&&& 1 & &&&&& 2 &&&& 3 \cr \+&& 2 && 2 &&& & 3&& 3 &&\cr \+&3 && 3 && 3 &&&& \cr }} \vskip10pt Notice that $_R Re_2 R=Re_2 \oplus Je_1$, and $Je_1 =Re_2 Je_1 =\Tr Je_1 Re_2 $. Since $c(Je_1 )=5$ and $c(Re_2 )=3$, we see that $Je_1$ (and hence $_R Re_2 R$) is not projective. Thus $R$ is not a left hereditary ring. We claim that $R$ is l-hereditary. To see this, it will suffice to show that any non-zero homomorphism $h:Re_2 \rightarrow Je_1 $ is monic. But any such homomorphism $h$ is given by right multiplication by some $x=e_2 xe_1 \in e_2 Je_1 =\langle a,b\rangle$. Thus $x=\alpha a +\beta b$ for some $\alpha ,\beta \in k$. Now, $Re_2 =\langle e_2 ,c,d\rangle$. Suppose $\lambda_1 e_2 +\lambda_2 c+\lambda_3 d \in \rm Ker\it (h)$, with $\lambda_1 ,\lambda_2 ,\lambda_3 \in k$. Then $0=h(\lambda_1 e_2 +\lambda_2 c +\lambda_3 d)= \lambda_1 \alpha a + \lambda_1 \beta b + \lambda_2 \alpha ca +(\lambda_3 \alpha +\lambda_2 \beta )da + \lambda_3 \beta db$. If either $\alpha \not= 0$ or $\beta \not= 0$, we see that $\lambda_i =0$ for $i=1,2,3$. Thus $h$ is monic and $R$ is l-hereditary. \vskip10pt The rings in Examples 1 and 3 are l-hereditary rings which are not left hereditary. We will show that under conditions weaker than those of Theorem 1, the notions of l-hereditary and left hereditary are the same. \vskip10pt \goodbreak \ub {Theorem 2}. Suppose that $J$ is a direct sum of local left ideals. Then the following are equivalent. \item {(a)} $R$ is left hereditary. \item {(b)} $R$ is l-hereditary. \item {(c)} Every primitive ideal of $R$ is an heredity ideal. \vskip10pt \ub {Proof}. (a $\Rightarrow$ b) This is clear. (b $\Rightarrow$ c) Let $R$ be l-hereditary and let $I=Re_i R$ be a primitive ideal of $R$. As we have seen, if $R$ is l-hereditary then $eJe=0$ for every primitive idempotent $e$. Thus it will suffice to show that $_R I$ is projective. For $\ds {j\not= i}$, we have $\ds {Re_i Re_j =} \Tr Re_j Re_i =Re_i Je_j$. Thus $$ Re_i R=\bigoplus_{j=1}^n Re_i Re_j =Re_i \oplus \bigoplus_{j\not= i} Re_i Je_j.$$ Now, each non-zero $Re_i Je_j$ is a direct sum of local left ideals of the form Im$(\rho_x )$, where the homomorphism $\rho_x :Re_i \rightarrow Je_j$ is given by right multiplication by $x$. By assumption, $\rho_x$ is monic and thus $Re_i R$ is projective. (c $\Rightarrow$ a) Assume that every primitive ideal is heredity. Recall that $R$ is left hereditary if and only if $Je_k$ is projective for each $k$, $\ds {1\leq k\leq n}$ (see [5] ex. 18.10). Let $1\leq k\leq n$. Since $J=\oplus_{i=1}^n Je_i$ is a direct sum of local left ideals, $\ds Je_k =\oplus_{j=1}^l L_j$ for some collection $\ds {L_1,\cdd ,L_l}$ of local left ideals. Let $1\leq i\leq l$ and consider the local left ideal $L_i$. Assume $L_i$ has projective cover $Re_t$. Notice that $\ds {t\not= k}$ since $\ds {Re_k Je_k =0}$. Since $Re_t R$ is heredity, $\ds {Re_t Re_k =Re_t Je_k}$ is a direct sum of copies of $Re_t$. But $\ds {Re_t Je_k =\oplus_{j=1}^l Re_t L_j}$ and hence $Re_t L_i =\Tr L_i Re_t =L_i$ is isomorphic (by Krull-Schmidt) to a copy of $Re_t$. Thus $L_i$ is projective and $R$ is left hereditary. \vskip10pt Since the radical of a left hereditary ring is necessarily a direct sum of local left ideals, we also have the following result. \vskip10pt \goodbreak \ub {Corollary 3}. Suppose $R$ is l-hereditary. Then $R$ is left hereditary if and only if $J$ is a direct sum of local left ideals. \vskip10pt Suppose $J$ is a direct sum of local left ideals. We then have the following dichotomy of the left global dimensions of such rings $R$ with the property that each primitive ideal is a projective left $R$-module. \vskip10pt \goodbreak \ub {Corollary 4}. If $J$ is direct sum of local left ideals and each primitive ideal is projective as a left $R$-module, then either $R$ is left hereditary or $\gd R =\infty$. \vskip10pt \ub {Proof}. If $R$ is not left hereditary, then by Theorem 2 there exists an $\ds {i,\ 1\leq i\leq n}$, such that $Re_i R$ is not an heredity ideal. Hence $\ds {e_i Je_i \not= 0}$ and by Corollary 1.5 of [6], we have that $\gd R =\infty$. \vskip10pt As noted above, if $ReR$ is an heredity ideal, then both $_R ReR$ and $ReR_R$ are projective. Thus there exists a right-hand version of Theorem 2 and we have the following result. \vskip10pt \goodbreak \ub {Corollary 5}. Suppose $_R J$ is a direct sum of local left ideals, and $J_R$ is a direct sum of local right ideals. Then $R$ is hereditary if and only if every primitive ideal of $R$ is an heredity ideal. \bigskip \vskip.5truein \centerline {\ub {\it References}} \vskip10pt \item {1.} \rm V. Dlab and C.M. Ringel, \it Quasi-hereditary algebras, \rm Illinois J. Math., 33 (1989), 280-291. \vskip10pt \item {2.} \rm \rm W.D. Burgess and K.R. Fuller, \it On quasihereditary rings, \rm Proc. Amer. Math. Soc., 106 (1989), 321-328. \vskip10pt \item {3.} \rm W.D. Burgess, K.R. Fuller, E.L. Green, D. Zacharia, \it Left monomial rings- a generalization of monomial algebras, \rm Osaka J. Math., 30 (1993), no. 3, 543-558. \vskip10pt \item {4.} \rm R. Martinez-Villa, \it Algebras stably equivalent to l-hereditary algebras, \rm Lecture Notes in Math. no. 832, Springer-Verlag, New York, 1980, 396-431. \vskip10pt \item {5.} \rm F.W. Anderson and K.R. Fuller, \it Rings and Categories of Modules, \rm \break Springer- Verlag, New York and Berlin, 2nd ed., 1992. \vskip10pt \item {6.} Darren D. Wick, \it A generalization of quasi-hereditary rings, \rm Comm. Algebra, 24 (1996), 1217-1227. \bye