direct product of rings
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From: Kusanagi on 2 Dec 2009 15:13 Why the direct product of two fields, considered as a ring is never itself a field? Thanks.
From: Arturo Magidin on 3 Dec 2009 01:21 On Dec 3, 12:13 am, Kusanagi <Kusan...(a)hotmail.com> wrote: > Why the direct product of two fields, considered as a ring is never itself a field? > > Thanks. What happens when you multiply (1,0) by (0,1)? Is either of them equal to the zero of the product? -- Arturo Magidin
From: Kusanagi on 2 Dec 2009 15:46 > On Dec 3, 12:13 am, Kusanagi <Kusan...(a)hotmail.com> > wrote: > > Why the direct product of two fields, considered as > a ring is never itself a field? > > > > Thanks. > > What happens when you multiply (1,0) by (0,1)? Is > either of them equal > to the zero of the product? > > -- > Arturo Magidin I see. How the tensor product of two fields, considered as a ring can get away with the above problem and become a field? Thanks.
From: Maarten Bergvelt on 3 Dec 2009 08:59 On 2009-12-03, Kusanagi <Kusanagi(a)hotmail.com> wrote: >> On Dec 3, 12:13 am, Kusanagi <Kusan...(a)hotmail.com> >> wrote: >> > Why the direct product of two fields, considered as >> a ring is never itself a field? >> > >> > Thanks. >> >> What happens when you multiply (1,0) by (0,1)? Is >> either of them equal >> to the zero of the product? >> > > I see. > > How the tensor product of two fields, considered as > a ring can get away with the above problem and become a field? What is the tensor product of 1 in the first field and 0 in the second field? -- Maarten Bergvelt
From: Hagen on 2 Dec 2009 23:54
> > On Dec 3, 12:13 am, Kusanagi > <Kusan...(a)hotmail.com> > > wrote: > > > Why the direct product of two fields, considered > as > > a ring is never itself a field? > > > > > > Thanks. > > > > What happens when you multiply (1,0) by (0,1)? Is > > either of them equal > > to the zero of the product? > > > > -- > > Arturo Magidin > > I see. > > How the tensor product of two fields, considered as > a ring can get away with the above problem and become > a field? > > Thanks. The tensor product over which ring or field? To form a tensor product of two commutative rings say, you have to consider them as algebras over a base ring R. In general the reason why the tensor product of two fields over some base can be a field is that the tensor product involves a lot of identification of pairs (you can contruct it using a certain equivalence relation on the product). H |