Localizations of a Ring at Localizable Sets, their Groups of Units and Saturations

In the paper, all rings and their homomorphisms are unital.

In [2], Ore’s method of localization was extended to localizable left Ore sets, a criterion was given of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore’s method of localization). Applications are given for certain classes of rings (semi-prime Goldie rings, Noetherian commutative rings, the algebras of polynomial integro-differential operators).

In [3], some basic results of the most general theory of one-sided fractions was developed and the following new concepts were introduced and studied: the almost Ore set, the localizable set, the perfect localizable set, the localization of a ring and a module at a localizable set. Their relations are given by the chain of inclusions:Localizable sets are generalization of Ore sets and denominator sets, and the localization of a ring/module at a localizable set is a generalization of localization of a ring/module at a denominator set.

In this paper, for a subset S of a ring R, the following concepts are introduced: the left saturation \(S^{sat}_l\), the right saturation \(S^{sat}_r\), the weak saturation \(S^{ws}\), the left weak saturation \(S^{ws}_l\), and the right weak saturation \(S^{ws}_r\). If the set S is a left or right localizable set then so are some of its saturations (Thereom 1.2).

In Sect. 2, for a left denominator set \(S\in \textrm{Den}_l(R)\) of a ring R, explicit descriptions of the group of units \((SR)^\times \) of the ring \(S^{-1}R\), and the monoids \((SR)_l^\times \) and \((SR)_r^\times \) of left and right invertible elements of \(S^{-1}R\) are obtained (Theorem 2.1).

The largest element \(S_l(R, \mathfrak {a}, S^{-1}R)\) in \((\textrm{Den}_l(R, \mathfrak {a}, S^{-1}R) \subseteq )\) and its characterizations where \(S\in \textrm{Den}_l(R)\). Let \(S,T\in \textrm{Den}_l(R)\). The denominator set T is called S-saturated if \(sr\in T\), for some \(s\in S\) and \(r\in R\), then \(r\in T\), and if \(r's'\in T\), for some \(s'\in S\) and \(r'\in R\), then \(r'\in T\), [2].

Let \(S\in \textrm{Den}_l(R)\), \(\mathfrak {a}=\textrm{ass}_l(S):=\{ r\in R\, | \, sr=0\) for some \(s\in S\}\), \(\textrm{Den}_l(R, \mathfrak {a}) := \{T\in \textrm{Den}_l(R)\, | \, \textrm{ass}_l(T)=\mathfrak {a}\}\), and[2, Proposition 3.1] describes the largest element \(S_l(R, \mathfrak {a}, S^{-1}R)\) of the partially ordered set \((\textrm{Den}_l(R, \) \(\mathfrak {a}, S^{-1}R), \subseteq )\). [2, Lemma3.3.(1)] gives another description of \(S_l(R, \mathfrak {a}, S^{-1}R)\) in terms of the group of units of the ring \(S^{-1}R\).

Similarly, for a right denominator set \(S\in \textrm{Den}_r (R, \mathfrak {a})\) (resp., \(S\in \textrm{Den}(R, \mathfrak {a})\)) , we denote by \(S_r (R, \mathfrak {a}, RS^{-1})\) (resp. \(S (R, \mathfrak {a}, S^{-1}R)\)) the largest element of the poset \((\textrm{Den}_r(R, \mathfrak {a}), \subseteq )\) (resp., \((\textrm{Den}(R, \mathfrak {a}), \subseteq )\)).

Definition. Let R be a ring and \(S\subseteq R\). The setsare called the left saturation, the right saturation, the weak saturation, the left weak saturation, and the right weak saturation of S, respectively. By the very definition,A ring R is called a finite ring if \(yx=1\) implies \(xy=1\) (every one-sided inverse is the inverse).

Theorem 1.2 is another characterization of the sets \(S_*(R, \mathfrak {a}, S^{-1}R)\) in terms of the five saturations above where \(*\in \{ l,r,\emptyset \}\). Its proof is given in Sect. 2.

Theorem 1.2 shows that saturations of \(*\) denominators sets are also \(*\) denominators sets.

Corollary 2.2 is a strengthening of Theorem 1.2 in the case when the ring \(S^{-1}R\) is either a domain or a one-sided Noetherian ring or does not contain an infinite direct sum of one-sided ideals. Corollary 2.3 shows that saturations are idempotent functors in the case of denominator sets.

Applications are given for the algebra \(\mathbb {S}_n\) of one-sided inverses (Proposition 2.5) where saturations are explicitly described (the algebra \(\mathbb {S}_n\) is not a finite ring. It is neither left nor right Noetherian, not a domain and contains infinite direct sums of left and right ideals).

Finiteness criterion for a localization of a ring via its saturations. Theorem 1.3 is a finiteness criterion for a localization of a ring at a localizable set which is given in terms saturations.

The largest element \(\mathcal {S}_*(R, \mathfrak {a}, S^{-1}R)\) in \((\mathbb {L}_*(R, \mathfrak {a}, S^{-1}R) \subseteq )\) and its characterizations where \(S\in \mathbb {L}_*(R, \mathfrak {a})\). In Sect. 3, the results of Sect. 2 for the denominator sets are generalized for localizable sets. At the beginning of Sect. 3, some results are collected from [3] on localizable sets and localizations of rings at localizable sets. Proposition 3.8.(2), is an explicit description of the largest element \( \mathcal {S}_* (R, \mathfrak {a}, S^{-1}A)\) of the partially ordered set \((\mathbb {L}_*(R, \mathfrak {a}, \mathcal {R}), \subseteq )\) of all \(*\) localizable sets S in R with \(\textrm{ass}_R(S)=\mathfrak {a}\) and \(R\langle S^{-1}\rangle \simeq \mathcal {R}\) where \(*\in \{ l,r,\emptyset \}\). Theorem 3.10 is another characterization of the set \(\mathcal {S}_*(R, \mathfrak {a}, R\langle S^{-1}\rangle )\) which is given in terms of the five saturations (it is an analogue of Theorem 1.2 but for localizable sets). In the case of Ore sets, we can strengthen Theorem 3.10, see Theorem 1.4.

Definition. Let R be a ring and \(S\subseteq R\). The setsare called the left bi-saturation, the right bi-saturation, the weak bi-saturation, the left weak bi-saturation, and the right weak bi-saturation of S, respectively. By the very definition,

The proof of Theorem 1.4 is given at the end of Sect. 3.

Let R be a ring. A multiplicative subset S of R is called a left Ore set if it satisfies the left Ore condition: for each \(r\in R\) and \(s\in S\), \( Sr\bigcap Rs\ne \emptyset \). Let \(\textrm{Ore}_l(R)\) be the set of all left Ore sets of R. For \(S\in \textrm{Ore}_l(R)\), \(\textrm{ass}_l (S) :=\{ r\in R\, | \, sr=0 \,\, \mathrm{for\; some}\,\, s\in S\}\) is an ideal of the ring R.

A left Ore set S is called a left denominator set of the ring R if \(rs=0\) for some elements \( r\in R\) and \(s\in S\) implies \(tr=0\) for some element \(t\in S\), i.e., \(r\in \textrm{ass}_l (S)\). Let \(\textrm{Den}_l(R)\) be the set of all left denominator sets of R. For \(S\in \textrm{Den}_l(R)\), let \(S^{-1}R=\{ s^{-1}r\, | \, s\in S, r\in R\}\) be the left localization of the ring R at S (the left quotient ring of R at S). Let us stress that in Ore’s method of localization one can localize precisely at left denominator sets. In a similar way, right Ore and right denominator sets are defined. Let \(\textrm{Ore}_r(R)\) and \(\textrm{Den}_r(R)\) be the set of all right Ore and right denominator sets of R, respectively. For \(S\in \textrm{Ore}_r(R)\), the set \(\textrm{ass}_r(S):=\{ r\in R\, | \, rs=0\) for some \(s\in S\}\) is an ideal of R. For \(S\in \textrm{Den}_r(R)\), \(RS^{-1}=\{ rs^{-1}\, | \, s\in S, r\in R\}\) is the right localization of the ring R at S.

Given ring homomorphisms \(\nu _A: R\rightarrow A\) and \(\nu _B :R\rightarrow B\). A ring homomorphism \(f:A\rightarrow B\) is called an R-homomorphism if \(\nu _B= f\nu _A\). A left and right set is called an Ore set. Let \(\textrm{Ore}(R)\) and \(\textrm{Den}(R)\) be the set of all Ore and denominator sets of R, respectively. For \(S\in \textrm{Den}(R)\),(an R-isomorphism) is the localization of the ring R at S, and \(\textrm{ass}(R):=\textrm{ass}_l(R) = \textrm{ass}_r(R)\).

For a ring R and \(*\in \{ l,r, \emptyset \}\), \(\textrm{Den}_*(R, 0)\) be the set of \(*\) denominator sets T of R such that \(T\subseteq \mathcal {C}_R\), i.e., the multiplicative set T is a \(*\) Ore set of R that consists of regular elements of the ring R.

The group of units \((S^{-1}R)^\times \)and monoids of one-sided inverses of a localization \(S^{-1}R\) where \(S\in \textrm{Den}_l(R)\). For a ring R, we denote by \(R^\times \) its group of units. Let \(R^\times _l:=\{ a\in R\, | \, ba=1\) for some \(b\in R\}\) and \(R^\times _r:=\{ a\in R\, | \, ab=1\) for some \(b\in R\}\), the sets of left and right invertible elements of the ring R, respectively. The sets \(R^\times _l\) and \(R^\times _r\) are multiplicative monoids that contain the group \(R^\times \) and \(R^\times = R^\times _l\cap R^\times _r\). The ring R is called a finite ring if \(ab=1\) implies \(ba=1\) (every one-sided inverse is the inverse). The ring R is a finite ring iff \(R^\times = R^\times _l= R^\times _r\) iff \(R^\times = R^\times _l\) iff \(R^\times = R^\times _r\). Every domain or a one-sided Noetherian ring is a finite ring. It is well-known that the algebra of one-sided inverses, \(\mathbb {S}_1=K\langle x,y\, | \, yx=1\rangle \), is not a finite ring (see [1] for generalizations and their properties). Let K be a field of characteristic zero and \(I_1\) be the subalgebra of the algebra \( \textrm{End }_K(K[x])\) which is generated by the K-derivation \(\partial =\frac{d}{dx}\) and the integration \(\int : K[x]\rightarrow K[x]\), \(x^n\mapsto \frac{x^n}{n!}\) where \(n\ge 0\). Then the K-algebra homomorphismis an isomorphism (since \(\partial \int =1\), \(\mathbb {S}_1=\bigoplus _{i,j\ge 0}Kx^iy^j\), and \(I_1=\bigoplus _{i,j\ge 0}K\int ^i\partial ^j\)). Clearly, the elements \(\partial \) and \(\int \) are not invertible (since \(\ker (\partial )=K\ne 0\)). Hence, the algebras \(\mathbb {S}_1\) and \(I_1\) are not finite.

For a ring R and its left denominator set S, Theorem 2.1 gives an explicit descriptions of the set \((S^{-1}R)^\times \), \((S^{-1}R)_l^\times \) and \((S^{-1}R)_r^\times \).

If the ring R is a domain the fact that \(\sigma ^{-1}((S^{-1}R)^\times )=\{ a\in R\, | \, ba\in S\}\) was proven in [4, Proposition 10].

The algebra \(\mathbb {S}_n\) of one-sided inverses. Let K be a field and \(K^\times \) be its group of units, and \(P_n:= K[x_1, \ldots , x_n]\) be a polynomial algebra over K.

Definition, [1]. The algebra \(\mathbb {S}_n\) of one-sided inverses of \(P_n\) is an algebra generated over a field K by 2n elements \(x_1, \ldots , x_n, y_1, \ldots , y_n\) that satisfy the defining relations:where \([a,b]:= ab-ba\), the commutator of elements a and b.

By the very definition, the algebra \(\mathbb {S}_n\) is obtained from the polynomial algebra \(P_n\) by adding commuting, left (or right) inverses of its canonical generators. The algebra \(\mathbb {S}_1\) is a well-known primitive algebra [5], p. 35, Example 2. Over the field \(\mathbb {C}\) of complex numbers, the completion of the algebra \(\mathbb {S}_1\) is the Toeplitz algebra which is the \(C^*\)-algebra generated by a unilateral shift on the Hilbert space \(l^2(\mathbb {N})\) (note that \(y_1=x_1^*\)). The Toeplitz algebra is the universal \(C^*\)-algebra generated by a proper isometry.

Clearly, \(\mathbb {S}_n=\mathbb {S}_1^{\otimes n}\) and \(\mathbb {S}_1= K\langle x,y\, | \, yx=1\rangle =\bigoplus _{i,j\ge 0}Kx^iy^j\). For each natural number \(d\ge 1\), let \(M_d(K):=\bigoplus _{i,j=0}^{d-1}KE_{ij}\) be the algebra of d-dimensional matrices where \(\{ E_{ij}\}\) are the matrix units, and \(M_\infty (K) := \varinjlim M_d(K)=\bigoplus _{i,j\in \mathbb {N}}KE_{ij}\) be the algebra (without 1) of infinite dimensional matrices. The algebra \(\mathbb {S}_1\) contains the ideal \(F:=\bigoplus _{i,j\in \mathbb {N}}KE_{ij}\), whereFor all natural numbers i, j, k, and l, \(E_{ij}E_{kl}=\delta _{jk}E_{il}\) where \(\delta _{jk}\) is the Kronecker delta function. The ideal F is an algebra (without 1) isomorphic to the algebra \(M_\infty (K)\) via \(E_{ij}\mapsto E_{ij}\). For all \(i,j\ge 0\),the direct sum of vector spaces. Thensince \(yx=1\), \(xy=1-E_{00}\) and \(E_{00}\in F\).

Lemma 2.4 is used in the proof of Proposition 2.5.

The algebra \(\mathbb {S}_n\) admits the involutioni.e. it is a K-algebra anti-isomorphism (\(\eta (ab) = \eta (b) \eta (a)\) for all \(a,b\in \mathbb {S}_n\)) such that \(\eta ^2 = \textrm{id}_{\mathbb {S}_n}\), the identity map on \(\mathbb {S}_n\). So, the algebra \(\mathbb {S}_n\) is self-dual (i.e. it is isomorphic to its opposite algebra, \(\eta : \mathbb {S}_n\simeq \mathbb {S}_n^{op}\)). This means that left and right algebraic properties of the algebra \(\mathbb {S}_n\) are the same.

Let \(\mathfrak {a}_n :=(x_1y_1-1, \ldots , x_ny_n-1)\), an ideal of \(\mathbb {S}_n\). By [1, Eq. (19)], the factor algebrais the Laurent polynomial algebra. Clearly, \(L_n^\times =\{ {\lambda } x^\alpha \, | \, {\lambda } \in K^\times , \alpha \in \mathbb {Z}^n\}\) where \(x^\alpha = x_1^{\alpha _1}\cdots x_n^{\alpha _n}\). LetThen \(\mathcal {L}_n:=\sigma ^{-1}(L_n^\times )=\{ {\lambda } x^\alpha +a\, | \, {\lambda } \in K^\times , \alpha \in \mathbb {Z}^n, a\in \mathfrak {a}_n\}\).

The goal of Sect. 3 is to generalize results of Sect. 2 for localizable sets. At the beginning of the section, we collect some results on localizable sets and localizations of rings at localizable sets from [3] that are used in the section.

The ring \( R\langle S^{-1}\rangle \). Let R be a ring and S be a multiplicative set in R (that is \(SS\subseteq S\), \(1\in S\) and \(0\not \in S\)). Let \(R\langle X_S\rangle \) be a ring freely generated by the ring R and a set \(X_S=\{ x_s\, | \, s\in S\}\) of free noncommutative indeterminates (indexed by the elements of the set S). Let us consider the factor ringof the ring \(R\langle X_S\rangle \) at the ideal \(I_S\) generated by the set of elements \(\{ sx_s-1, x_ss-1 \, | \, s\in S\}\).

The kernel of the ring homomorphismis denoted by \(\textrm{ass}(S) = \textrm{ass}_R(S)\). The ideal \(\textrm{ass}_R(S)\) of R has a complex structure, its description is given in [3, Proposition 2.12] when S is a left localizable set.

Definition, [3]. A multiplicative set S of a ring R is called a left localizable set of R ifwhere \(\overline{R}= R/ \mathfrak {a}\), \(\mathfrak {a}= \textrm{ass}_R(S)\) and \(\overline{S}= (S+\mathfrak {a}) / \mathfrak {a}\), i.e., every element of the ring \( R\langle S^{-1}\rangle \) is a left fraction \(\overline{s}^{-1} \overline{r}\) for some elements \(\overline{s}\in \overline{S}\) and \( \overline{r}\in \overline{R}\). Similarly, a multiplicative set S of a ring R is called a right localizable set of R ifi.e., every element of the ring \( R\langle S^{-1}\rangle \) is a right fraction \( \overline{r}\overline{s}^{-1}\) for some elements \(\overline{s}\in \overline{S}\) and \( \overline{r}\in \overline{R}\). A right and left localizable set of R is called a localizable set of R.

The sets of left localizable, right localizable and localizable sets of R are denoted by \( \mathbb {L}_l(R)\), \( \mathbb {L}_r(R)\) and \( \mathbb {L}(R)\), respectively. Clearly, \( \mathbb {L}(R)= \mathbb {L}_l(R)\cap \mathbb {L}_r(R)\). In order to work with these three sets simultaneously we use the following notation \( \mathbb {L}_*(R)\) where \(*\in \{ l,r, \emptyset \}\) and \(\emptyset \) is the empty set \((\mathbb {L}(R) = \mathbb {L}_\emptyset (R))\). LetFor an ideal \(\mathfrak {a}\) of R, let \( \mathbb {L}_*(R, \mathfrak {a})=\{ S\in \mathbb {L}_*(R)\, | \, \textrm{ass}_R(S) = \mathfrak {a}\}\). Thenis a disjoint union of non-empty sets.

The ideals \(\mathfrak {a}(S)\), \({}'\mathfrak {a}(S)\) and \(\mathfrak {a}'(S)\). For each element \(r\in R\), let \(r\cdot : R\rightarrow R\), \(x\mapsto rx\) and \(\cdot r : R\rightarrow R\), \(x\mapsto xr\). The setsare called the sets of left and right regular elements of R, respectively. Their intersectionis the set of regular elements of R. The ringsare called the classical left and right quotient rings of R, respectively. Goldie’s Theorem states that the ring \(Q_{l, cl}(R)\) is a semisimple Artinian ring iff the ring R is semiprime, \(\textrm{udim}(R)<\infty \) and the ring R satisfies the a.c.c. on left annihilators (\(\textrm{udim}\) stands for the uniform dimension).

We have the inclusionwhere \(*\in \{ l,r,\emptyset \}\), [3, Lemma 1.2]. The proof of Proposition 3.1 contains an explicit description of the ideal \(\mathfrak {a}(S)\). The ideal \(\mathfrak {a}(S)\) is the key part in the definition of perfect localizable sets that are introduced in [3].

The structure of the ring \( R\langle S^{-1}\rangle \) and its universal property. Recall that for a ring R, we denote by \(R^\times \) its group of units. Theorem 3.2 describes the structure and the universal property of the ring \( R\langle S^{-1}\rangle \).

In view of Theorem 3.2.(1,2), for \(S\in \mathbb {L}_*(R)\) we denote by \(S^{-1}R\) the ring \( R\langle S^{-1}\rangle \) for \(*\in \{ l,\emptyset \}\) and by \(RS^{-1}\) for \(*\in \{ r,\emptyset \}\). In particular, for \(S\in \mathbb {L}(R)\),Elements of the rings \(S^{-1}R\) and \( RS^{-1}\) are denoted by \(s^{-1}r\) and \(rs^{-1}\), respectively, where \(s\in S\) and \(r\in R\). Sometime, in order to make arguments shorter for \(S\in \mathbb {L}_r(R)\) we denote the ring \(RS^{-1}\) by \(S^{-1}R\).

For the algebra \(\mathbb {S}_n\) and its multiplicative set Y, Lemma 3.3 presents explicitly all the ingredients of Proposition 3.1 and Theorem 3.2.

For the algebra \(\mathbb {S}_n\) and its multiplicative set X, Lemma 3.4 presents explicitly all the ingredients of Proposition 3.1 and Theorem 3.2.

For each element \(\alpha = (\alpha _1, \ldots , \alpha _n)\in \mathbb {N}^n\), letRecall that \(\sigma : \mathbb {S}_n \rightarrow \mathbb {S}_n/\mathfrak {a}_n=L_n, \;\; a\mapsto a+\mathfrak {a}_n\). Notice thatConsider a subgroup of units \(\tilde{Z}:=\{ x^\alpha \, | \, \alpha \in \mathbb {Z}^n\}\) of the algebra \(L_n\). Its pre-imageis a submonoid of \((\mathbb {S}_n, \cdot )\).

The posets \((\mathbb {L}_*(R), \subseteq )\) and \((\mathbb {L}\textrm{oc}_*(R),\rightarrow )\). The set \((\mathbb {L}\textrm{oc}_*(R, \mathfrak {a}), \rightarrow )\) is a poset where \(A_1\rightarrow A_2\) if \(A_1 = R\langle S_1^{-1}\rangle \) and \(A_2= R\langle S_2^{-1}\rangle \) for some localizable sets \(S_1, S_2 \in \mathbb {L}_*(R, \mathfrak {a})\) such that the map \(A_1\rightarrow A_2\), \(s_1^{-1}r\mapsto s_1^{-1}r\) if \(*\in \{ l,\emptyset \}\) (resp., \(rs_1^{-1}\mapsto rs_1^{-1}\) if \(*=r\)) is a well-defined homomorphism. Moreover, enlarging if necessary the denominator set \(S_2\) we can assume that \(S_1\subseteq S_2\) (for example, by taking \(S_2= \sigma _2^{-1}(A_2^\times )\) where \(\sigma _2: R\rightarrow A_2\), \(r\mapsto \frac{r}{1}\), see Proposition  3.8.(2)). By Proposition  3.8.(2),In the same way, the poset \((\mathbb {L}\textrm{oc}_*(R), \rightarrow )\) is defined, i.e. \(A_1\rightarrow A_2\) if there exist \(S_1, S_2\in \mathbb {L}_*(R)\) such that \(S_1\subseteq S_2\), \(A_1 = R\langle S_1^{-1}\rangle \) and \(A_2= R\langle S_2^{-1}\rangle \), \(A_1\rightarrow A_2\) stands for the map \(\varphi :A_1\rightarrow A_2\), \(s_1^{-1}r\mapsto s_1^{-1}r\) if \(*\in \{ l,\emptyset \}\) (resp., \(rs_1^{-1}\mapsto rs_1^{-1}\) if \(*=r\)). The mapis an epimorphism from the poset \(( \mathbb {L}_*(R), \subseteq )\) to \((\mathbb {L}\textrm{oc}_l(R), \rightarrow )\). For each ideal \(\mathfrak {a}\in \textrm{Ass}_*(R)\), it induces the epimorphism of the posets \((\mathbb {L}_*(R, \mathfrak {a}), \subseteq )\) and \((\mathbb {L}\textrm{oc}_*(R, \mathfrak {a}), \rightarrow )\),The sets \(\mathbb {L}_*(R)\) and \(\mathbb {L}\textrm{oc}_*(R)\) are the disjoint unionsFor each ideal \(\mathfrak {a}\in \textrm{Ass}_*(R)\), the set \(\mathbb {L}_*(R, \mathfrak {a})\) is the disjoint unionwhere \(\mathbb {L}_*(R, \mathfrak {a}, A ):=\{ S\in \mathbb {L}_*(R, \mathfrak {a})\, | \, R\langle S^{-1}\rangle \simeq A\), an R-isomorphism\(\}\).

The largest element \(\mathcal {S}_*(R, \mathfrak {a}, S^{-1}R)\) in \((\mathbb {L}_*(R, \mathfrak {a}, S^{-1}R) \subseteq )\) and its characterizations where \(S\in \mathbb {L}_*(R, \mathfrak {a})\). Proposition 3.7.(1) is a practical criterion for a multiplicative set S of a ring R to belong to the set \(\mathbb {L}_*(R, \mathfrak {a})\).

Proposition 3.8.(2), is an explicit description of the largest element \( \mathcal {S}_* (R, \mathfrak {a}, S^{-1}A)\) of the partially ordered set \((\mathbb {L}_*(R, \mathfrak {a}, S^{-1}R), \subseteq )\).

By Lemma 3.5,For the algebra \(\mathbb {S}_n\) and its multiplicative set \(S=Z\) (see (3.7)), Lemma 3.9 gives an explicit description of the set \(S_*(\mathbb {S}_n, \mathfrak {a}_n, L_n)\), see Proposition 3.8.(2).

Theorem 3.10 is another characterization of the set \(\mathcal {S}_*(R, \mathfrak {a}, R\langle S^{-1}\rangle )\) in terms of the five saturations.

By [3, Theorem 1.6.(1,2)], if \(S\in \textrm{Ore}(R)\) then \(S\in \mathbb {L}(R)\) and