Citizenship question

If a man that is a U.S. citizen and also a citizen of Ireland,gets married to a Russian woman,what country would it be better for her to apply to for citizenship?

I don't know the immigration laws of Ireland, so difficult to answer. I do know it is a tough road to hull if she wishes to immigrate to the US not imposable but difficult. I would suggest you contact the department of Immigration in both countries and put the question to them. It is not difficult to contact these departments and this is there jobs so you will get a real answer rather than a guess or an opinion.
Good luck

Ireland is an EU country. USA is what we (in Denmark) call a third country.
As an EU citizen or as a wife/husband of an EU citizen, you will have special rights in the EU countries.
A person from a country outside the union will have a lot of problems in order to obtain a citizenship.
There are special rules concerning marriages. Often it is much easier, if the person from the third country (Russia or USA) gets a permit to work in an EU country.
If a person works in Sweden or Germany (quite liberal immigration laws) - he or she will much easier get access to Denmark (with very restrictful laws).
- A piece of advice: Check the immigration laws carefully, and use the differences so that you can benefit from it.
An example: A person works for three months in Sweden. After this period it is quite easy to move to Denmark.
-You can find special groups on the Internet that can give you some (legal) tips that might be very handy..

AJ-

A lady moving to the USA is not required to be a "citizen" to obtain permanent residency. She can apply for a "permanent resident card" here and never be a citizen.

Your new wife can apply for a "Permanent Residence Card". But it is still considered "temporary" until she meets time living in the US requirements for true permanent residency.

I have been through this process with my former wife (My EX) who is a Swiss national. My children have two passports, Swiss and US. My former wife lived in the US for nearly 15 years before she decided to actually apply for US Citizenship because it was not a requirement for her to remain in the US.

Here is the actual ISCIS Link:

http://www.uscis.gov/portal/site/uscis/menuitem.5af9bb95919f35e66f614176543f6d1a/?vgnextoid=ddb2194d3e88d010VgnVCM10000048f3d6a1RCRD&vgnextchannel=4f719c7755cb9010VgnVCM10000045f3d6a1RCRD