When is it appropriate to report interactions? I have two categorical variables, A and B. Each categorical variable has three levels(0,1,2). There is a certain dependent variable P against which I make a plot and see that there is an interaction between A and B. In my next step, I make a model when I regress upon P with A*B (model_interaction). When I look at the summary of this model I see that certain interactions terms are significant. Here is my question, is this enough evidence to say that there are significant interactions? 
Why am I asking this? 
Along with the interaction model, I also made a linear model with A and B regressed on P (model_linear). When I compared model_interaction and model_linear I found no statistical difference between the two and I also found that the AIC score for model_linear was lower. So, after I've seen all of this do I still say that I have found significant interactions? 
Just to summarize: 
model_linear: P ~ A + B
model_interaction: P ~ A * B
Evidence for interaction:
1) Plots showing clear interaction.
2) Model with the interaction terms have significant p-values
Evidence against it:
1) Interaction model not significantly different from linear model
2) Linear model has lower AIC score compared to the interaction model. 
Do I say there are interactions or not?
 A: I think you made a programming mistake. When comparing two lms that are the same except for the inclusion of an interaction term, anova should give the same $p$-value as summary gives for the interaction term. For example:
> coef(summary(lm(area ~ peri + shape * perm, data = rock)))["shape:perm",]
    Estimate   Std. Error      t value     Pr(>|t|) 
-10.71598140   4.84042362  -2.21385198   0.03219246 
> anova(lm(area ~ peri + shape + perm, data = rock), lm(area ~ peri + shape * perm, data = rock))
Analysis of Variance Table

Model 1: area ~ peri + shape + perm
Model 2: area ~ peri + shape * perm
  Res.Df      RSS Df Sum of Sq      F  Pr(>F)  
1     44 74326644                              
2     43 66721703  1   7604941 4.9011 0.03219 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

See how the number 0.03219 appears in both outputs?
anova, at least for the case of two lms, uses an $F$-test, not a $χ^2$ test, and it is indeed equivalent to the $t$-test used by summary.lm.
As for AIC, you said:

The fact that these two [models] are not significantly different does not tell us which model is better than the other. To check which is better I was using the AIC scores to compare.

It's true that a non-significant result is uninformative, but it doesn't make sense to follow that up with AIC. The only reason to use a $p$-value for model selection is if you believe a priori that the simpler model is better and you'll stick with it unless the more complex model provides "enough" of an increase in fit. So if you're using the significance-testing approach, and you don't get a significant result, you should use the simpler model; end of story. AIC is a different approach with different standards of how to choose a model. Use the significance test or AIC, but not both. Of these, AIC is more sophisticated and is likely to be better for any real-world purpose.
