in a class gave me the concept of initial and terminal object. I started to look for this objects in different categories. I already proved that $\mathbb{Z}$ is initial and the zero-ring is terminal in the category of rings, but I can't find the initial and terminal objects (if they exists) in the category of local rings, so any help would be appreciated. Thanks

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    $\begingroup$ It's relatively easy to show that there aren't any. Just think about the characteristics of the rings and/or their residue fields. $\endgroup$ – Johannes Hahn May 9 '16 at 18:20

A correct but somewhat uninteresting answer (as pointed out by Johannes Hahn) is that there is no initial local ring, in the vanilla category of local rings and local homomorphisms. As Johannes said, this is more or less for the same reason that the category of fields has no initial object and no terminal object. In more detail, if $(R, m)$ is a local ring, then pick a field $k$ whose characteristic differs from that of the field $R/m$. Then there can be no local homomorphism $(R, m) \to (k, 0)$, so there can be no initial object. There is no terminal object either.

Things become rather more interesting however if one changes the formulation to talk not about local homomorphisms between local rings in $\text{Set}$, but about local homomorphisms between local rings living in (possibly) different toposes. What could this mean? (To the OP: it sounds as if you are just beginning a study of category theory, so not all of this answer may be immediately understandable for you at the moment. It is partly for the wider community.)

For full details, I think I'll refer you to Myles Tierney's classic paper On the Spectrum of a Ringed Topos in the 1976 collection Algebra, Topology, and Category Theory (Academic Press). You can see a bit of it in Google Books. In brief, though, one can define local ring objects $(R, U)$ in a topos $E$ (now switching attention to the group of units $U$ which classically is complementary to the maximal ideal). If $(R, U)$ is a local ring object in $E$ and $(S, V)$ is a local ring object in $F$, then define a map between them to be a left exact left adjoint $f^\ast: E \to F$ together with a local homomorphism $\phi: f^\ast R \to S$. We in fact have a 2-category consisting of locally ringed toposes, homomorphisms $(f^\ast, \phi)$, and natural transformations $f^\ast \to g^\ast$ making an evident triangle commute.

As shown by Tierney, it turns out that there is a locally ringed topos which is 'initial', or to be more technically accurate, 2-initial in this 2-category. It is called the big Zariski topos of the integers $\mathbb{Z}$. This interpretation was an important advance in the general theory of classifying toposes. This is also connected with Monique Hakim's thesis (Topos annelés et schémas relatifs), which interprets the étale topos of a ring as a classifying topos for strict local rings.

Related material may be found in these MO discussions: Joyal's construction of the spectrum of a commutative ring and What does an etale topos classify?

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