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edited Aug 3 2010 at 10:54 |
I'd like to add some obvious remarks, as a non-specialist (please in case correct or rectify me). My impression is that the matter is important, but not substantial.
Non-unital commutative rings and non-unital homomorphisms do exist in the mathematical world, but nevertheless a non-unital ring $A$ may always be seen as an ideal of a unital ring, via the usual construction $\eta_A:A\to\mathbb{Z}\times A$, and a non-unital homomorphism may be seen as the restriction of a unital homomorphism. In some case, there is also a "concrete" embedding : for instance one can embed the non-unital convolution algebra of $L^1(\mathbb{R}^n)$ in the distributions. However, I do have in mind convolution equations in $L^1(\mathbb{R}^n)$ were the computations are greatly simplified just by introducing an even formal unity.
In other words, one loses nothing in generality by stating : commutative rings have a unity, and homomorphism preserve it, as books of commutative algebra usually do. So the situation is similar to the assumption of completeness for metric spaces: we may, and often do consider, a metric space as a dense subset of a complete metric space. The analogy is made precise by the category language: the category of unital commutative rings is a non-full, reflective subcategory of the category of non-unital ring with non-unital homomorphisms (non=non necessarily); the unity of the corresponding adjunction is of course given by the above homomorphisms $\eta_A$ (the co-unity being $\epsilon_R:\mathbb{Z}\times R\ni (n,x)\mapsto n\mathbf{1}_R+x\in R$ for unital rings $R$).
There are of course other situations in mathematics were one prefers not to profit of a canonical embedding; certainly (almost) nobody would see $\mathbb {N}$ as a sub-monoid of $\mathbb{R}$. Clearly these choices depend on practicality, and partially on fashion.
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edited Aug 3 2010 at 10:48 |
I'd like to add some obvious remarks, as a non-specialist (please in case correct or rectify me). My impression is that the matter is important, but not substantial.
Non-unital commutative rings and non-unital homomorphisms do exist in the mathematical world, but a non-unital ring $A$ may always be seen as an ideal of a unital ring, via the usual construction $\eta_A:A\to\mathbb{Z}\times A$, and a non-unital homomorphism may be seen as the restriction of a unital homomorphism. In some case, there is also a "concrete" embedding : for instance one can embed the non-unital convolution algebra of $L^1(\mathbb{R}^n)$ in the distributions. However, I do have in mind convolution equations in $L^1(\mathbb{R}^n)$ were the computations are greatly simplified just by introducing an even formal unity.
In other words, one loses nothing in generality by stating : commutative rings have a unity, and homomorphism preserve it, as books of commutative algebra usually do. So the situation is similar to the assumption of completeness for metric spaces: we may, and often do consider, a metric space as a dense subset of a complete metric space. The analogy , of course, is made precise by the category language: the category of unital commutative rings is a non-full, reflective subcategory of the category of non-unital ring with non-unital homomorphisms (non=non necessarily); the unity of the corresponding adjunction is of course given by the above homomorphisms $\eta_A$ (the co-unity being of course $\epsilon_R:\mathbb{Z}\times R\ni (n,x)\mapsto n\mathbf{1}_R+x\in R$ for unital rings $R$).
There are of course other situations in mathematics were one prefers not to profit of a canonical embedding; certainly (almost) nobody would see $\mathbb {N}$ as a sub-module sub-monoid of $\mathbb{R}$. Clearly these choices depend on praticitypracticality, and partially on fashion.
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edited Aug 3 2010 at 10:40 |
I'd like to add some obvious remarks, as a non-specialist (please in case correct or rectify me). My impression is that the matter is important, but not substantial.
Non-unital commutative rings and non-unital homomorphisms do exist in the mathematical world, but a non-unital ring $A$ may always be seen as an ideal of a unital ring, via the usual construction $\eta_A:A\to\mathbb{Z}\times A$, and a non-unital homomorphism may be seen as the restriction of a unital homomorphism. In some case, there is also a "concrete" embedding : for instance one can embed the non-unital convolution algebra of $L^1(\mathbb{R}^n)$ in the distributions. However, I do have in mind convolution equations in $L^1(\mathbb{R}^n)$ were the computations are greatly simplified just by introducing an even formal unity.
In other words, one loses nothing in generality by stating : commutative rings have a unity, and homomorphism preserve it, as books of commutative algebra usually do. So the situation is similar to the assumption of completenes completeness for metric spaces: we may, and often do consider, a metric space as a dense subset of a complete metric space. The analogy, of course, is made precise by the category language: the category of unital commutative rings is a non-full, reflective subcategory of the category of non-unital ring with non-unital homomorphisms (non=non necessarily); the unity of the corresponding adjunction is of course given by the above homomorphisms $\eta_A$ (the co-unity being of course $\epsilon_R:\mathbb{Z}\times R\ni (n,x)\mapsto n\mathbf{1}_R+x\in R$ for unital rings $R$).
There are of course other situations in mathematics were on one prefers not to profit of a canonical embedding; certainly (almost) nobody would see $\mathbb {N}$ as a sub-module of $\mathbb{R}$. Clearly these choices depend on praticity, and partially on fashion.
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edited Aug 3 2010 at 10:27 |
I'd like to add some obvious remarks, as a non-specialist (please in case correct or rectify me). My impression is that the matter is important, but not substantial.
Non-unital commutative rings and non-unital homomorphisms do exist in the mathematical world, but a non-unital ring $A$ may always be seen as an ideal of a unital ring, via the usual construction $\eta_A:A\to\mathbb{Z}\times A$, and a non-unital homomorphism may be seen as the restriction of a unital homomorphism. In some case, there is also a "concrete" embedding : for instance one can embed the non-unital convolution algebra of $L^1(\mathbb{R}^n)$ in the distributions. However, I do have in mind convolution equations in $L^1(\mathbb{R}^n)$ were the computations are greatly simplified just by introducing an even formal unity.
In other words, one loses nothing in generality by stating : commutative rings have a unity, and homomorphism preserve it, as books of commutative algebra usually do. So the situation is similar to the assumption of completenes for metric spaces: we may, and often do consider, a metric space as a dense subset of a complete metric space. The analogy, of course, is made precise by the category language: the category of unital commutative rings is a non-full, reflective subcategory of the non unital category of non-unital ring with non unital non-unital homomorphisms (non=non necessarily); the unity of the corresponding adjunction is of course given by the above homomorphisms $\eta_A$ .(the co-unity being of course $\epsilon_R:\mathbb{Z}\times R\ni (n,x)\mapsto n\mathbf{1}_R+x\in R$ for unital rings $R$).
There are of course other situations in mathematics were on prefers not to profit of a canonical embedding; certainly (almost) nobody would see $\mathbb {N}$ as a sub-module of $\mathbb{R}$. Clearly these choices depend on praticity, and partially on fashion.
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edited Aug 3 2010 at 10:13 |
I'd like to add some obvious remarks, as a non-specialist (please in case correct or rectify me). My impression is that the matter is important, but not substantial.
Non-unital commutative rings and non-unital homomorphisms do exist in the mathematical world, but a non-unital ring $A$ may alwais always be seen as an ideal of a unital ring, via the usual construction $\eta_A:A\to\mathbb{Z}\times A$, and a non-unital homomorphism may be seen as the restriction of a unital homomorphism. In some case, there is also natural a "concrete" embedding : for instance one can embed the non-unital convolution algebra of $L^1(\mathbb{R}^n)$ in the distributions. However, and I do have in mind convolution equations in $L^1(\mathbb{R}^n)$ were the computations are greatly simplified just by introducing an even formal unity.
In other words, one loses nothing in generality by stating : commutative rings have a unity, and homomorphism preserve it, as books of commutative algebra usually do. So the situation is similar to the assumption of completenes for metric spaces: we may, and often do consider, a metric space as a dense subset of a complete metric space. The analogy, of course, is made precise by the category language: the category of unital commutative rings is a non-full, reflective subcategory of the non unital ring with non unital homomorphisms (non=non necessarily); the unity of the corresponding adjunction is of course given by the above homomorphisms $\eta_A$.\eta_A$ .
There are of course other situations in mathematics were on prefers not to profit of a canonical embedding; certainly (almost) nobody would see $\mathbb {N}$ as a sub-module of $\mathbb{R}$. Clearly these choices depend on praticity, and partially on fashion.
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answered Aug 3 2010 at 10:03 |
I'd like to add some obvious remarks, as a non-specialist (please in case correct or rectify me). My impression is that the matter is important, but not substantial.
Non-unital commutative rings and non-unital homomorphisms do exist in the mathematical world, but a non-unital ring $A$ may alwais be seen as an ideal of a unital ring, via the usual construction $\eta_A:A\to\mathbb{Z}\times A$, and a non-unital homomorphism may be seen as the restriction of a unital homomorphism. In some case, there is also natural embedding: for instance one can embed the non-unital convolution algebra of $L^1(\mathbb{R}^n)$ in the distributions, and I do have in mind convolution equations in $L^1(\mathbb{R}^n)$ were the computations are greatly simplified just by introducing an even formal unity.
In other words, one loses nothing in generality by stating : commutative rings have a unity, and homomorphism preserve it, as books of commutative algebra usually do. So the situation is similar to the assumption of completenes for metric spaces: we may, and often do consider, a metric space as a dense subset of a complete metric space. The analogy, of course, is made precise by the category language: the category of unital commutative rings is a non-full, reflective subcategory of the non unital ring with non unital homomorphisms (non=non necessarily); the unity of the corresponding adjunction is of course given by the above homomorphisms $\eta_A$.
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